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Chapter 13: Q 15. (page 1014)

Let g1(x)andg2(x)be two continuous functions such that g1(x)鈮�g2(x)on the interval [a,b], and let be the region in the xy-plane bounded by g1andg2on[a,b]. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of 惟. How is this iterated integral related

to the definite integral you would have used to compute the area of 惟in Chapter 4?

Short Answer

Expert verified

The iterated integral that gives area of region is鈭�abgg1(x)鈭�2g2(x)dydx

Step by step solution

01

Given Information

Two continuous functions g1(x)andg2(x)such that g1(x)<g2(x)in a,b.

Region 惟is bounded by curves g1andg2in a,b.

02

Consideration

The region bounded on left by x=aand right by x=a, y=g1(x)below and y=g1(x)above is of type I

Hence, surface integral becomes

鈭�惟dA=鈭�abgg1(x)鈭�2g2(x)dydx

Integral that gives area of region is 鈭�ab鈭�g1(x)g2(x)dydx.

鈭�惟dA=鈭�ab[y]g1(x)g2(x)dx

=鈭�abg2(x)-g1(x)dx

Hence, integral鈭�惟dAgives area of region惟

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Most popular questions from this chapter

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

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

鈭�-33鈭�-9-x29-x2鈭�-33f(x,y,z)dzdydx

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)}?

Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

f(x,y)=xyWhereR={(x,y)|-2鈮�x鈮�3,-1鈮�y鈮�5}

In Exercises, let S=(x,y)鈭�x2+y2鈮�1andx鈮�0.

If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of S about the x-axis, the y-axis, and the origin.
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