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Chapter 13: Q. 25 (page 1055)

Evaluate the iterated integral :

∫01 ∫24 ∫−15 x+yz2dxdydz

Short Answer

Expert verified

36

Step by step solution

01

Step 1. Given information.

Given integral is :

∫01 ∫24 ∫−15 x+yz2dxdydz

We have to evaluate it.

02

Step 2. Evaluate.

=∫01 ∫24 ∫−15 x+yz2dxdydzl=∫01 ∫24 ∫−15 xdx+∫−15 yz2dxdydz=∫01 ∫24 x22−15+yz2(x)−15dydz=∫01 ∫24 522−(−1)22+yz2(5−(−1))dydz=∫01 ∫24 12+6yz2dydz

03

Step 3. Integrate with respect to y.

Integrate with respect to y

=∫01 ∫24 12dy+6z2∫24 yd⋅ydz=∫01 12(y)24+6z2y2224dz=∫01 12(4−2)+6z2422−222dz=∫01 24+36z2dz

Integrate with respect to z

=24∫01 dz+36∫01 z2dzAlgebra of=24(z)01+36z3301Integ=24(1−0)+3613−0Apply lin=24+12=36∴∫01 ∫24 ∫−15 x+yz2dxdydz=36

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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" Ωyz={(y,z)|a≤y≤bandh1(y)≤z≤h2(y)}be a set of points in the yz-plane, and let localid="1650354983053" Ω={(x,y,z)|(y,z)∈Ωyzandg1(y,z)≤x≤g2(y,z)}be a set of points in 3-space. Find an iterated triple integral equal to the triple integral localid="1650353288865" ∭Ωf(x,y,z)dV. How would your answer change iflocalid="1650352747263" Ωyz={(y,z)|a≤z≤bandh1(z)≤x≤h2(z)}?

In Exercises 57–60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.

Assume that the density at each point in Ris proportional to the distance of the point from the xy-plane.

(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are x¯=2andy¯=32,respectively.

(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.

In the following lamina, all angles are right angles and the density is constant:

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∫01 ∫06−y ∫03−3y−(1/2)z (y−z)dxdzdy

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