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Chapter 13: Q. 1 (page 1025)

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral

12鈭�0蟺cos23胃d胃

Short Answer

Expert verified

The area is蟺4square units.

Step by step solution

01

Step 1. Given information

Integral:

12鈭�0蟺cos23胃d胃
02

Step 2. Plot the curve

Since the area of a function r=f(胃)is 12鈭�0蟺r2d胃

So by comparing the given integral we get,

r=cos3胃

So the graph of this curve is:

03

Step 3. Evaluate integral

12鈭�0蟺cos23胃d胃=12鈭�0蟺cos23胃d胃=12鈭�0蟺1+cos6胃2d胃=14鈭�0蟺1+cos6胃d胃=14胃+sin6胃60蟺=14蟺+sin6蟺6-0=蟺4

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Most popular questions from this chapter

In Exercises 45鈥�52, rewrite the indicated integral with the specified order of integration.

Exercise 41 with the order dy dx dz.

Evaluate the iterated integral :

鈭�01鈥�鈭�03鈥�鈭�0sin鈭�1鈦�x鈥�(x+y)dzdydx

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

鈭�鈭�RxsinxcosydA,whereR={(x,y)|鈭�3鈮�x鈮�2and鈭�2鈮�y鈮�2}

Evaluate the sums in Exercises 23鈥�28.

鈭�j=1m鈭�k=1n(j-k)

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.
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