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Multivariable Calculus 11th Edition

Multivariable Calculus (11th Edition)

Book Edition11th Edition
Author(s)Larson
ISBN9781337275378
PublisherCengage Learning
SubjectMath
Chapter 14, Section 14.1, Exercises, Exercise 1
Page 976

Explain what is meant by an iterated integral. How is it evaluated?

Explanation

*To give you a brief description, iterated integral is just an integration of more than 1 variable, f(x, y). This is commonly used when getting an area using integration. It is evaluated by integrating first the inner portion of the equation. So here is the process on how to conduct iterated integration:


First, recall the formula of iterated integration


f(x,y)dA=y1y2x1x2f(x,y)dxdy


For this formula, we will first integrate the "inner" part of the equation.


y1y2[x1x2f(x,y)dx]dy


As you can see above, the function will be integrated with respect to x. This means that x is the variable and we will treat y as constant.


Second, evaluate the inner part of the equation.


y1y2[f(x,y)]x1x2dy


Third, after evaluating the inner part with respect to x, it is now time to integrate the function with respect to y. In this, x now serves as the constant while y is the variable.


*Let us have an example for us to understand how iterated integration is being done.


0213x2ydxdy


In this problem, we need to integrate the inner part of the function first. Let us separate it for us to visualize more.


0213x2ydxdy=02[13x2ydx]dy


Integrate now with respect to x, treating y as constant.


02[13x2ydx]dy=02[3x3y]13dy = 02[333y313y]dy=02[326y]dy = 32602[y]dy


Next, integrate with respect to y.


32602ydy=326[2y2]02=326(222202)=352

Answer

Iterated integral is just an integration of more than 1 variable, f(x, y). This is commonly used when getting an area using integration. It is evaluated by integrating first the inner portion of the equation.

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