Area between Curves - Revision history

Lila at 16:41, 29 October 2021

2021-10-29T16:41:55Z

Lila: /* Area between two curves */

2021-10-29T16:41:12Z

Area between two curves

Lila: /* Resources */

2021-10-06T18:47:05Z

Resources

Lila at 18:42, 6 October 2021

2021-10-06T18:42:39Z

Glenn.lahodny at 14:13, 4 December 2020

2020-12-04T14:13:28Z

Glenn.lahodny at 14:12, 4 December 2020

2020-12-04T14:12:36Z

Glenn.lahodny at 13:50, 4 December 2020

2020-12-04T13:50:43Z

James.kercheville: Added video links

2020-08-19T19:49:10Z

Added video links

New page

[https://youtu.be/DRFyNHdVgUA Finding Areas Between Curves] by patrickJMT

[https://youtu.be/1yxSA-FZ0gk Areas Between Curves - 2 Regions] by patrickJMT

[https://youtu.be/70NQ3ISYihw Area Between Curves - Integrating with Respect to y] by patrickJMT

[https://youtu.be/iAvF1Rn7How Finding Area by Integrating with Respect to Y] by patrickJMT

[https://youtu.be/Xne6Hv9useE Area Between Curves - Integrating with Respect to y - Part 2] by patrickJMT

[https://youtu.be/8w9l99W8ucY Area of a Region Bounded by 3 Curves (Calculus)] by patrickJMT

[https://youtu.be/AeInlCa-1Q8 Area between curves - dx] by Krista King

[https://youtu.be/fGvSy6U209Y Area between curves - dy] by Krista King

[https://youtu.be/7lrn-8FiTW0 Area between curves - sketching] by Krista King

[https://youtu.be/ltxdcegn8xc Area Between Two Curves] by The Organic Chemistry Tutor

[https://youtu.be/wXMr2YYDMiM Area Between Two Curves & Under Curve - Respect to Y & X - Calculus & Integration] by The Organic Chemistry Tutor

@@ Line 26: / Line 26: @@
 :<math>A=\int\limits_c^d \bigl(f(y)-g(y)\bigr)dy</math> .
 Regardless of the form of the functions, we basically use the same formula.
 [[File:Closed path integral defined.png|Closed_path_integral_defined]]
@@ Line 83: / Line 84: @@
 [https://youtu.be/wXMr2YYDMiM Area Between Two Curves & Under Curve - Respect to y & x] by The Organic Chemistry Tutor

@@ Line 26: / Line 26: @@
 :<math>A=\int\limits_c^d \bigl(f(y)-g(y)\bigr)dy</math> .
 Regardless of the form of the functions, we basically use the same formula.
+[[File:Closed path integral defined.png|Closed_path_integral_defined]]
 ==Resources==

← Older revision		Revision as of 18:47, 6 October 2021
Line 29:		Line 29:

	==Resources==		==Resources==
		+	* [https://en.wikibooks.org/w/index.php?title=Calculus/Area&stable=1 Area], WikiBooks: Calculus
		+
		+	===Videos===
	[https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U		[https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U

← Older revision		Revision as of 18:42, 6 October 2021
Line 1:		Line 1:
		+	==Introduction==
		+	Finding the area between two curves, usually given by two explicit functions, is often useful in calculus.
		+
		+	In general the rule for finding the area between two curves is
		+
		+	<math>A=A_{\rm top}-A_{\rm bottom}</math> or
		+
		+	If f(x) is the upper function and g(x) is the lower function
		+
		+	<math>A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx</math>
		+
		+	This is true whether the functions are in the first quadrant or not.
		+
		+	==Area between two curves==
		+	Suppose we are given two functions <math>y_1=f(x)</math> and <math>y_2=g(x)</math> and we want to find the area between them on the interval <math>[a,b]</math> . Also assume that <math>f(x)\ge g(x)</math> for all <math>x</math> on the interval <math>[a,b]</math> . Begin by partitioning the interval <math>[a,b]</math> into <math>n</math> equal subintervals each having a length of <math>\Delta x=\frac{b-a}{n}</math> . Next choose any point in each subinterval, <math>x_i^</math> . Now we can 'create' rectangles on each interval. At the point <math>x_i</math> , the height of each rectangle is <math>f(x_i^)-g(x_i^)</math> and the width is <math>\Delta x</math> . Thus the area of each rectangle is <math>\bigl[f(x_i^)-g(x_i^)\bigr]\Delta x</math> . An ''approximation'' of the area, <math>A</math> , between the two curves is
		+	:<math>A:=\sum_{i=1}^n \Big[f(x_i^)-g(x_i^)\Big]\Delta x</math> .
		+	Now we take the limit as <math>n</math> approaches infinity and get
		+	:<math>A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(x_i^)-g(x_i^)\Big]\Delta x</math>
		+	which gives the exact area. Recalling the definition of the definite integral we notice that
		+	:<math>A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx</math> .
		+	This formula of finding the area between two curves is sometimes known as applying integration with respect to the ''x''-axis since the rectangles used to approximate the area have their bases lying parallel to the ''x''-axis. It will be most useful when the two functions are of the form <math>y_1=f(x)</math> and <math>y_2=g(x)</math> . Sometimes however, one may find it simpler to integrate with respect to the ''y''-axis. This occurs when integrating with respect to the ''x''-axis would result in more than one integral to be evaluated. These functions take the form <math>x_1=f(y)</math> and <math>x_2=g(y)</math> on the interval <math>[c,d]</math> . Note that <math>[c,d]</math> are values of <math>y</math> . The derivation of this case is completely identical. Similar to before, we will assume that <math>f(y)\ge g(y)</math> for all <math>y</math> on <math>[c,d]</math> . Now, as before we can divide the interval into <math>n</math> subintervals and create rectangles to approximate the area between <math>f(y)</math> and <math>g(y)</math> . It may be useful to picture each rectangle having their 'width', <math>\Delta y</math> , parallel to the ''y''-axis and 'height', <math>f(y_i^)-g(y_i^)</math> at the point <math>y_i^*</math>, parallel to the ''x''-axis. Following from the work above we may reason that an ''approximation'' of the area, <math>A</math> , between the two curves is
		+	:<math>A:=\sum_{i=1}^n \Big[f(y_i^)-g(y_i^)\Big]\Delta y</math> .
		+	As before, we take the limit as <math>n</math> approaches infinity to arrive at
		+	:<math>A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(y_i^)-g(y_i^)\Big]\Delta y</math> ,
		+	which is nothing more than a definite integral, so
		+	:<math>A=\int\limits_c^d \bigl(f(y)-g(y)\bigr)dy</math> .
		+	Regardless of the form of the functions, we basically use the same formula.
		+
		+
		+	==Resources==
	[https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U		[https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U

@@ Line 1: / Line 1: @@
-* [https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U
+[https://www.youtube.com/watch?v=de-0e0nYf5I Determining Area Between Two Curves] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=E_1aDcOoDtE The Area Between Two Graphs] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=q7De3oOp4Ug Ex 1: Area Between a Linear and Quadratic Function (respect to x)] by James Sousa, Math is Power 4U
+[https://www.youtube.com/watch?v=E_1aDcOoDtE The Area Between Two Graphs] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=n2JqTztMrvQ Ex 2: Area Between a Linear and Exponential Function (respect to x)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=OqMbRAEQ-cg Ex 3: Area Between Two Exponential Functions (respect to x)] by James Sousa, Math is Power 4U
+[https://www.youtube.com/watch?v=q7De3oOp4Ug Ex 1: Area Between a Linear and Quadratic Function (respect to x)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=58Esio4q4jk Ex 4: Area Between Two Quadratic Functions (respect to x)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=CPhUolqwLNk Ex 1: Area Bounded by Two Functions] by James Sousa, Math is Power 4U
+[https://www.youtube.com/watch?v=n2JqTztMrvQ Ex 2: Area Between a Linear and Exponential Function (respect to x)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=lskEXsJaJV0 Ex 2: Area Bounded by Two Functions (Two Regions)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=BH-V6Gx2n8E Ex 3: Area Bounded by Two Trig. Functions] by James Sousa, Math is Power 4U
+[https://www.youtube.com/watch?v=OqMbRAEQ-cg Ex 3: Area Between Two Exponential Functions (respect to x)] by James Sousa, Math is Power 4U
-* [https://www.youtube.com/watch?v=N4fEuEQLG5c Ex: Determine a Function Given the Area Between Two Functions] by James Sousa, Math is Power 4U
 [https://youtu.be/DRFyNHdVgUA Finding Areas Between Curves] by patrickJMT

@@ Line 10: / Line 10: @@
 * [https://www.youtube.com/watch?v=N4fEuEQLG5c Ex: Determine a Function Given the Area Between Two Functions] by James Sousa, Math is Power 4U
-[https://youtu.be/DRFyNHdVgUA Finding Areas Between Curves]  by patrickJMT
+[https://youtu.be/DRFyNHdVgUA Finding Areas Between Curves] by patrickJMT
-[https://youtu.be/AeInlCa-1Q8 Area between curves - dx]  by Krista King
+[https://youtu.be/1yxSA-FZ0gk Areas Between Curves - 2 Regions] by patrickJMT
-[https://youtu.be/fGvSy6U209Y Area between curves - dy]  by Krista King
+[https://youtu.be/iAvF1Rn7How Finding Area by Integrating with Respect to y] by patrickJMT
-[https://youtu.be/7lrn-8FiTW0 Area between curves - sketching]  by Krista King
+[https://youtu.be/70NQ3ISYihw Area Between Curves - Integrating with Respect to y] by patrickJMT
-[https://youtu.be/ltxdcegn8xc Area Between Two Curves]  by The Organic Chemistry Tutor
+[https://youtu.be/Xne6Hv9useE Area Between Curves - Integrating with Respect to y - Part 2] by patrickJMT
-[https://youtu.be/wXMr2YYDMiM Area Between Two Curves & Under Curve - Respect to Y & X - Calculus & Integration]  by The Organic Chemistry Tutor
+[https://www.youtube.com/watch?v=M2G68iQzYQE Finding Areas Between Curve: Region and Functions Given] by patrickJMT