   Chapter 15, Problem 19RE

Chapter
Section
Textbook Problem

Calculate the iterated integral by first reversing the order of integration. ∫ 0 1 ∫ x 1 cos ( y 2 )   d y   d x

To determine

To calculate: The iterated integral by reversing the order of integration.

Explanation

Given:

The iterated integral is 01x1cos(y2)dydx .

Calculation:

Since x is varying between 0 and 1 and y varies between x and 1, the region lies between two lines y=x and y=1 . From this, it is concluded that the region is the triangle for which the sides are y=x y=1 and x=0 . Thus, the limits of x and y can also be intimated as follows. x varies from 0 to y and y varies from 0 to 1. Then, the given iterated integral becomes,

01x1cos(y2)dydx=010ycos(y2)dxdy

First, compute the integral with respect to x.

010ycos(y2)dxdy=01[0ycos(y2)dx]dy=01[ycos(y2)]0ydy

Apply the limit values for x,

010ycos(y2)dxdy=01[((y)cos(y2))((

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