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Chapter 13: Q. 63 (page 991)

Use a double integral to prove that the area of the circle with radius R and equationr=2Rcos胃is蟺R2.

Short Answer

Expert verified

The area of the circle is

A=蟺R2P

Step by step solution

01

Given information

The objective of this problem is to use double integral to prove that the area of the circle with radius R and equation r=2Rcos胃is蟺R2.

02

calculation

Draw the circle

Plot of r=2Rcos胃

Given circle is symmetrical about the horizontal axis. Therefore area of circle in polar form can be expressed as the twice of area of upper half circle.

A=2鈭�6a1鈭�5xrdrd胃

Here, 胃1=0,胃2=蟺2andr1=0,r2=r

A=2鈭�0蟺/2鈭�0r-2蟺cos胃rdrd胃

Integrate with respect to r first

A=2鈭�0*/2r2202蟺ened胃鈭�xndx=xn+1n+1+C

A=2鈭�0蟺/2(2Rcos胃)2-02

A=2R2鈭�0蟺/22cos2胃d胃A=2R2鈭�0蟺/2[1+cos2胃]d胃

Integrate with respect to胃

A=2R2胃+12sin2胃0x/2鈭�cosxdx=sinx+CA=2R2蟺2+12sin蟺-0

localid="1650471847418" A=蟺R2

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Most popular questions from this chapter

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

鈭�鈭�Rsin(x+2y)dAWhereR={(x,y)|0鈮�x鈮�蟺,0鈮�y鈮�蟺2}

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

鈭�-33鈭�-9-z29-z2鈭�03-x2+z2f(x,y,z)dydxdz

State Fubini's theorem.

Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each (xj*,yk*) is the midpoint of the subrectangle

Rjk={(x,y)鈭�xj-1鈮�xj*鈮�xjandyk-1鈮�yk*鈮�yk}.

Refer to your answer to Exercise 10 or to Definition 13.3.

Evaluate the iterated integral :

鈭�01鈥�鈭�06鈭�y鈥�鈭�03鈭�3y鈭�(1/2)z鈥�(y鈭�z)dxdzdy
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