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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.

-2204-x20ydzdydx

Short Answer

Expert verified

-2204-x20ydzdydx=163

Step by step solution

01

Given

We have to find the volume by using the triple integral.

-2204-x20ydzdydx
02

Evaluate the integral

V=-2204-x20ydzdydxV=-2204-x2ydydxV=-22y2204-x2dxV=12-22(4-x2)dxV=2×1202(4-x2)dxV=4x-x3302V=8-83V=163cubicunits

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Most popular questions from this chapter

Let f(x,y,z)be a continuous function of three variables, let Ωxy={(x,y)|axbandh1(x)yh2(x)}be a set of points in the xy-plane, and let Ω={(x,y,z)|(x,y)Ωxyandg1(x,y)zg2(x,y)}be a set of points in 3-space. Find an iterated triple integral equal to the the triple integralΩf(x,y,z)dV. How would your answer change ifΩxy={(x,y)|aybandh1(y)xh2(y)}?

Find the masses of the solids described in Exercises 53–56.

The solid bounded above by the paraboloid with equation z=8-x2-y2and bounded below by the rectangle R={(x,y,0)|1x2and0y2}in the xy-plane if the density at each point is proportional to the square of the distance of the point from the origin.

Explain how to construct a midpoint Riemann sum for a function of three variables over a rectangular solid for which each xi*,yj*,zk*is the midpoint of the subsolid role="math" localid="1650346869585" Rijk={x,y,k|xi-1xi*xi,yj-1yj*yjandzk-1zk*zk}. Refer either to your answer to Exercise 4or to Definition 13.14.

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}.

Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Discuss the similarities and differences between the definition of the double integral found in Section 13.1and the definition of the triple integral found in this section.

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