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Chapter 13: Q. 35 (page 1015)

Find the volume of the solid bounded above by the given function over the specified region

f(x,y)=10鈭�2x+y, with 惟 the region from Exercise 21

Short Answer

Expert verified

The volume is :

10e+e24-494cubic units.

Step by step solution

01

Step 1. Given information

The function:

f(x,y)=10-2x+y

The region in execise 21

02

Step 2. To setup integral to find volume

In the given region we can see that

0鈮�y鈮�ex0鈮�x鈮�1

So volume of the function over this region is:

V=鈭�01鈭�0exf(x,y)dydx=鈭�01鈭�0ex10-2x+ydydx

03

Step 3. To evaluate integral.

First integrate the function with respect to y.

鈭�0ex10-2x+ydy=10y-2xy+y220ex=10ex-2xex+e2x2-0=10ex-2xex+e2x2

Now integrate with respect to x.

鈭�0110ex-2xex+e2x2dx=10ex-2xex-ex+e2x401=10e-2(e-e)+e24-10-2(0-1)+14=10e-0+e24-10-2-14=10e+e24-494

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Most popular questions from this chapter

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

鈭�惟鈥�z蚁(x,y,z)dV

Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

Evaluate the iterated integral :

鈭�01鈥�鈭�03鈥�鈭�0sin鈭�1鈦�x鈥�(x+y)dzdydx

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

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

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.
See all solutions

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