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Chapter 13: Q 61. (page 1014)

Evaluate the double integral over the specified region.

鈭�惟x2y3dAwhere 惟 is region in first quadrant bounded by curvesy=x2andx=y2

Short Answer

Expert verified

The integral is3110.

Step by step solution

01

Given Information

We are given with 鈭�惟x2y3dA

The curves arey=x2andx=y2.

02

Graph

The region in graph is shown below:

03

Simplification

The region of integration is region between y=x2andy=x.

For type I integral

0鈮�x鈮�1,x2鈮�y鈮�x

Hence, integral is simplified as

鈭�01鈭�x2xx2y3dydx=鈭�01y44x2xx2dx

=14鈭�01x2-x8x2dx

=14鈭�01x4-x10dx

=14x55-x111101

=1415-111

=3110

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Most popular questions from this chapter

Let f(x,y,z)be a continuous function of three variables, let localid="1650352548375" 惟yz={(y,z)|a鈮�y鈮�bandh1(y)鈮�z鈮�h2(y)}be a set of points in the yz-plane, and let localid="1650354983053" 惟={(x,y,z)|(y,z)鈭�惟yzandg1(y,z)鈮�x鈮�g2(y,z)}be a set of points in 3-space. Find an iterated triple integral equal to the triple integral localid="1650353288865" 鈭�惟f(x,y,z)dV. How would your answer change iflocalid="1650352747263" 惟yz={(y,z)|a鈮�z鈮�bandh1(z)鈮�x鈮�h2(z)}?

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

鈭�鈭�RxexydA,where{R=(x,y)|0鈮�x鈮�1and0鈮�y鈮�ln5}

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The first-octant solid bounded by the coordinate planes and the plane 3x + 4y + 6z = 12 if the density at each point is proportional to the distance of the point from the xz-plane.

Evaluate the iterated integral :

鈭�01鈥�鈭�03鈥�鈭�0sin鈭�1鈦�x鈥�(x+y)dzdydx
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