Arc Length and Surface Area

Arc Length

Suppose that we are given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} that is continuous on an interval $[a,b]$ and we want to calculate the length of the curve drawn out by the graph of $f(x)$ from $x=a$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=b} . If the graph were a straight line this would be easy — the formula for the length of the line is given by Pythagoras' theorem. And if the graph were a piecewise linear function we can calculate the length by adding up the length of each piece.

The problem is that most graphs are not linear. Nevertheless we can estimate the length of the curve by approximating it with straight lines. Suppose the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is given by the formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} for $a\leq x\leq b$ . We divide the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} subintervals with equal width $\Delta x$ and endpoints Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0,x_1,\ldots,x_n} . Now let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i=f(x_i)} so $P_{i}=(x_{i},y_{i})$ is the point on the curve above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} . The length of the straight line between $P_{i}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{i+1}} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl|P_iP_{i+1}\bigr|=\sqrt{(y_{i+1}-y_i)^2+(x_{i+1}-x_i)^2}}

So an estimate of the length of the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is the sum

$\sum _{i=0}^{n-1}{\bigl |}P_{i}P_{i+1}{\bigr |}$

As we divide the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} into more pieces this gives a better estimate for the length of $C$ . In fact we make that a definition.

Length of a Curve'

The length of the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\le x\le b} is defined to be

$L=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\bigl |}P_{i+1}P_{i}{\bigr |}$

The Arclength Formula

Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} . Then the length of the curve given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\int\limits_a^b \sqrt{1+f'(x)^2}dx}

And in Leibniz notation

$L=\int \limits _{a}^{b}{\sqrt {1+\left({\tfrac {dy}{dx}}\right)^{2}}}dx$

Proof: Consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i+1}-y_i=f(x_{i+1})-f(x_i)} . By the Mean Value Theorem there is a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_i} in $(x_{i+1},x_{i})$ such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i+1}-y_i=f(x_{i+1})-f(x_i)=f'(z_i)(x_{i+1}-x_i)}

So

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl|P_iP_{i+1}\bigr|}	$={\sqrt {(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}}}$
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sqrt{(x_{i+1}-x_i)^2+f'(z_i)^2(x_{i+1}-x_i)^2}}
	$={\sqrt {{\bigl (}1+f'(z_{i})^{2}{\bigr )}(x_{i+1}-x_{i})^{2}}}$
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sqrt{1+f'(z_i)^2}\Delta x}

Putting this into the definition of the length of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} gives

$L=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\sqrt {1+f'(z_{i})^{2}}}\Delta x$

Now this is the definition of the integral of the function $g(x)={\sqrt {1+f'(x)^{2}}}$ between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} (notice that $g$ is continuous because we are assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'} is continuous). Hence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\int\limits_a^b \sqrt{1+f'(x)^2}dx}

as claimed.

Example

Length of the curve $y=2x$ from $x=0$ to $x=1$}}

As a sanity check of our formula, let's calculate the length of the "curve" $y=2x$ from $x=0$ to $x=1$ . First let's find the answer using the Pythagorean Theorem.

$P_{0}=(0,0)$

and

$P_{1}=(1,2)$

so the length of the curve, $s$ , is

$s={\sqrt {2^{2}+1^{2}}}={\sqrt {5}}$

Now let's use the formula

$s=\int \limits _{0}^{1}{\sqrt {1+\left({\tfrac {d(2x)}{dx}}\right)^{2}}}\,dx=\int \limits _{0}^{1}{\sqrt {1+2^{2}}}\,dx={\sqrt {5}}x{\bigg |}_{0}^{1}={\sqrt {5}}$

Exercises

1. Find the length of the curve $y=x{\sqrt {x}}$ from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} .

2. Find the length of the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\frac{e^x+e^{-x}}{2}} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} to $x=1$.

Arclength of a parametric curve

For a parametric curve, that is, a curve defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=f(t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=g(t)} , the formula is slightly different:

$L=\int \limits _{a}^{b}{\sqrt {f'(t)^{2}+g'(t)^{2}}}\,dt$

Proof: The proof is analogous to the previous one: Consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i+1}-y_i=g(t_{i+1})-g(t_i)} and $x_{i+1}-x_{i}=f(t_{i+1})-f(t_{i})$ .

By the Mean Value Theorem there are points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i} in $(t_{i+1},t_{i})$ such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i+1}-y_i=g(t_{i+1})-g(t_i)=g'(c_i)(t_{i+1}-t_i)}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i+1}-x_i=f(t_{i+1})-f(t_i)=f'(d_i)(t_{i+1}-t_i)}

So

${\bigl |}P_{i}P_{i+1}{\bigr |}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2}}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sqrt{f'(d_i)^2(t_{i+1}-t_i)^2+g'(c_i)^2(t_{i+1}-t_i)^2}}
	$={\sqrt {{\bigl (}f'(d_{i})^{2}+g'(c_{i})^{2}{\bigr )}(t_{i+1}-t_{i})^{2}}}$
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sqrt{f'(d_i)^2+g'(c_i)^2}\Delta t}

Putting this into the definition of the length of the curve gives

$L=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\sqrt {f'(d_{i})^{2}+g'(c_{i})^{2}}}\Delta t$

This is equivalent to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\int\limits_a^b \sqrt{f'(t)^2+g'(t)^2}\,dt}

Exercises

3. Find the circumference of the circle given by the parametric equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)=R\cos(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t)=R\sin(t)} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} running from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} .

4. Find the length of one arch of the cycloid given by the parametric equations $x(t)=R{\bigl (}t-\sin(t){\bigr )}$ , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t)=R\bigl(1-\cos(t)\bigr)} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} running from $0$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} .

Exercise Solutions

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{13\sqrt{13}-8}{27}}
2. ${\frac {e-{\frac {1}{e}}}{2}}$
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi R}
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8R}

Surface Area

Suppose we are given a function $f$ and we want to calculate the surface area of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.

If the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a straight line, other methods such as surface area formulae for cylinders and conical frusta can be used. However, if $f$ is not linear, an integration technique must be used.

Recall the formula for the lateral surface area of a conical frustum:

$A=2\pi rl$

where $r$ is the average radius and $l$ is the slant height of the frustum.

For $y=f(x)$ and $a\leq x\leq b$ , we divide $[a,b]$ into subintervals with equal width $\delta x$ and endpoints $x_{0},x_{1},\ldots ,x_{n}$ . We map each point $y_{i}=f(x_{i})$ to a conical frustum of width Δx and lateral surface area $A_{i}$ .

We can estimate the surface area of revolution with the sum

$A=\sum _{i=0}^{n}A_{i}$

As we divide $[a,b]$ into smaller and smaller pieces, the estimate gives a better value for the surface area.

Definition (Surface of Revolution)

The surface area of revolution of the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} about a line for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\le x\le b} is defined to be

$A=\lim _{n\to \infty }\sum _{i=0}^{n}A_{i}$

The Surface Area Formula

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a continuous function on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and $r(x)$ represents the distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} to the axis of rotation. Then the lateral surface area of revolution about a line is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = 2\pi\int_a^b r(x) \sqrt{1+f'(x)^2} \, dx}

And in Leibniz notation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=2\pi\int_a^b r(x) \sqrt{1 + \left(\tfrac{dy}{dx}\right)^2}\,dx}

Proof:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	$=\lim _{n\to \infty }\sum _{i=1}^{n}A_{i}$
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\sum_{i=1}^n 2\pi r_il_i}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =2\pi\cdot\lim_{n\to\infty}\sum_{i=1}^n r_il_i}

As $n\to \infty$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x\to 0} , we know two things:

1. the average radius of each conical frustum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_i} approaches a single value
2. the slant height of each conical frustum $l_{i}$ equals an infitesmal segment of arc length

From the arc length formula discussed in the previous section, we know that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_i=\sqrt{1+f'(x_i)^2}}

Therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =2\pi\cdot\lim_{n\to\infty}\sum_{i=1}^n r_il_i}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =2\pi\cdot\lim_{n\to\infty}\sum_{i=1}^n r_i\sqrt{1+f'(x_i)^2}\Delta x}

Because of the definition of an integral $\int _{a}^{b}f(x)dx=\lim _{n\to \infty }\sum _{i=1}^{n}f(c_{i})\Delta x_{i}$ , we can simplify the sigma operation to an integral.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=2\pi\int_a^b r(x) \sqrt{1+f'(x)^2} dx}

Or if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is in terms of $y$ on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [c,d]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=2\pi\int_c^d r(y) \sqrt{1+f'(y)^2} dy}

Resources

• Arc Length, WikiBooks: Calculus
• Surface Area, WikiBooks: Calculus

Arc Length

• Arc Length - Part 1 of 2 by James Sousa, Math is Power 4U
• Arc Length - Part 2 of 2 by James Sousa, Math is Power 4U
• Ex: Find the Arc Length of a Linear Function by James Sousa, Math is Power 4U
• Ex: Find the Arc Length of a Radical Function by James Sousa, Math is Power 4U
• Ex: Find the Arc Length of a Quadratic Function by James Sousa, Math is Power 4U

• Deriving the Arc Length Formula in Calculus by patrickJMT
• Arc Length by patrickJMT

• Arc Length y=f(x) by Krista King
• Arc length x=g(y) by Krista King

• Arc Length Intro by Khan Academy
• Arc Length Example by Khan Academy
• Arc Length Example by Khan Academy

• Arc Length by The Organic Chemistry Tutor

Surface Area

• Surface Area of Revolution - Part 1 of 2 by James Sousa, Math is Power 4U
• Surface Area of Revolution - Part 2 of 2 by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Linear Function by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Sine Function by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Cubic Function About x-axis by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Square Root Function About x-axis by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Quadratic Function About y-axis by James Sousa, Math is Power 4U
• Ex: Surface Area of Revolution - Cube Root Function About y-axis by James Sousa, Math is Power 4U

• Finding Surface Area - Part 1 by patrickJMT
• Finding Surface Area - Part 2 by patrickJMT

• Surface Area of Revolution Example 1 by Krista King
• Surface Area of Revolution Example 2 by Krista King
• Surface Area of Revolution Example 3 by Krista King

• Surface Area of Revolution By Integration by The Organic Chemistry Tutor