Iterated Integrals and Fubini's Theorem

Iterated Integrals

We are about to look at a method of evaluating double integrals over rectangles and more general domains without using the definition of a double integral, but before we do so, we will first need to learn what an iterated integral.

Suppose that $z=f(x,y)$ is a two variable real-valued function, and suppose that $f$ is integrable for $a\leq x\leq b$ and $c\leq y\leq d$, that is $f$ is integrable over the rectangle:

${\begin{aligned}\quad R=\{(x,y)\in \mathbb {R} ^{2}:a\leq x\leq b,c\leq y\leq d\}=[a,b]\times [c,d]\end{aligned}}$

Just like partial differentiation with respect to a specific variable, we can also partial integrate with respect to a specific variable. For example, $\int _{c}^{d}f(x,y)\,dy$ means that we integrate $f$ with respect to $y$ while holding the variable $x$ as fixed. When we evaluate this integral, we will obtain a function in terms of $x$ only, and hence, we could then integrate the result from $a$ to $b$ with respect 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} as:

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 \begin{align} \quad \int_a^b \int_c^d f(x, y) \, dy \, dx \end{align}}

The result above is what we call an iterated integral which we will define below.

Definition: 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 z = f(x, y)} is a two variable real valued function, and 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 integrable over the rectangle 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 = [a, b] \times [c, d]} , then the Iterated Integral 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 f} over 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} 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 \int_a^b \int_c^d f(x, y) \, dy \, dx} .

We will see later that iterated integrals need not be over rectangles but instead can be done over more general domains.

One important property about iterated integrals is that we can partially integrate 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, y)} with respect to either variable 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} or 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} first, and then continue onward with integrating with respect to the second variable, that 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 \begin{align} \quad \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy \end{align}}

However, it is important to note that sometimes partial integrating with respect to a certain variable first will be a much easier process. Let's now look at some examples of evaluating iterated integrals.

Example 1

Evaluate the following iterated 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 \int_2^4 \int_1^3 x^3 + xy^2 \, dy \, dx} . Over what rectangle 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 f(x, y) = x^3 + xy^2} being integrated?

When evaluated iterated integrals over rectangles, we always want to work from the inside out. Let's first evaluate the inside integral with respect 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 y} while holding 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} as fixed.

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 \begin{align} \quad \int_1^3 x^3 + xy^2 \, dy =\left [ x^3y + \frac{xy^3}{3} \right ]_1^3 = \left ( 3x^3 + 9x \right ) - \left ( x^3 + \frac{x}{3} \right ) = 2x^3 + \frac{26x}{3} \end{align}}

Therefore we have 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 \begin{align} \quad \int_2^4 \int_1^3 x^3 + xy^2 \, dy \, dx = \int_2^4 2x^3 + \frac{26x}{3} \, dx \end{align}}

Evaluating this definite integral and we get 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 \begin{align} \quad \int_2^4 2x^3 + \frac{26x}{3} \, dx = \left [ \frac{x^4}{2} + \frac{26x^2}{6} \right ]_2^4 = \left ( 128 + \frac{416}{6} \right ) - \left ( 8 + \frac{104}{26} \right ) = 120 + 52 = 172 \end{align}}

Therefore we have 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 \begin{align} \quad \int_2^4 \int_1^3 x^3 + xy^2 \, dy \, dx = 172 \end{align}}

In this particular example, we are integrating over the rectangle 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 = [2, 4] \times [1, 3]} .

Example 2

Evaluate the following iterated 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 \int_0^{\pi} \int_{0}^{\pi} \sin x \cos y \, dx \, dy} . Over what rectangle 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 f(x, y) = \sin x \cos y} being integrated?

We will first start by evaluating the inner integral while holding 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} as fixed:

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 \begin{align} \quad \int_{\frac{\pi}{2}}^{\pi} \sin x \cos y \, dx = \left [ -\cos x \cos y \right ]_{0}^{\pi} = \left ( \cos y \right ) - \left ( -\cos y \right ) = 2\cos y \end{align}}

Therefore we have 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 \begin{align} \quad \int_0^{\pi} \int_{0}^{\pi} \sin x \cos y \, dx \, dy = \int_0^{\pi} 2 \cos y \, dy \end{align}}

Evaluating this definite integral and we get 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 \begin{align} \quad \int_0^{\pi} 2 \cos y \, dy = \left [ 2 \sin y \right]_0^{\pi} = 0 \end{align}}

In this particular example, we are integrating over the square 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 = [0, \pi] \times [0, \pi]} .

Resources

• Iterated Integrals, Paul's Online Notes

Licensing

Content obtained and/or adapted from:

• Iterated integrals, mathonline.wikidot.com under a CC BY-SA license
• Fubini's Theorem and Evaluating Double Integrals over Rectangles, mathonline.wikidot.com under a CC BY-SA license
• Fubini's Theorem for Evaluating Triple Integrals over Boxes, mathonline.wikidot.com under a CC BY-SA license