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Chapter 13: Q. 20 (page 1082)

Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.

鈭�02蟺鈭�01+sin胃rdrd胃

Short Answer

Expert verified

鈭�02蟺鈭�01+sin胃rdrd胃=123蟺-2

Step by step solution

01

Draw the region

The region determined by the limits is shown below,

02

Evaluate the integral

I=鈭�02蟺鈭�01+sin胃rdrd胃I=鈭�02蟺r2201+sin胃d胃I=12鈭�02蟺1+sin胃2d胃I=12鈭�02蟺1+sin2胃+2sin胃d胃I=12鈭�02蟺1+1-cos2胃2+2sin胃d胃I=12胃+12胃-14sin2胃-2cos胃02蟺I=122蟺+蟺-2I=123蟺-2

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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

Use Definition 13.4to evaluate the double integrals in Exercises 29鈥�32.

鈭�Rxy3dA

whereR={(x,y)鈭�-2鈮�x鈮�2&-1鈮�y鈮�1}

In Exercises 61鈥�64, let R be the rectangular solid defined by

R=(x,y,z)|0鈮�a1鈮�x鈮�a2,0鈮�b1鈮�y鈮�b2,0鈮�c2鈮�z鈮�c2.

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is a1+a22,b1+b22,c1+c22.

(b) Verify that a1+a22,b1+b22,c1+c22is the center of mass by using the appropriate integral expressions.

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

鈭�02鈭�03鈭�4x/34f(x,y,z)dzdxdy

Evaluate the sums in Exercises 23鈥�28.

鈭�j=1m鈭�k=1n(j-k)
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