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Chapter 10: Q. 39 (page 777)

Areas of regions bounded by polar functions: Find the areas of the following regions. The area that is inside both lemniscatesr2=cos2胃andr2=sin2胃

Short Answer

Expert verified

The area that is inside both lemniscates is1-12units

Step by step solution

01

Given information

r2=cos2胃andr2=sin2胃

02

Area

Find the limits of integration as,

sin2胃=cos2胃鈬�tan2胃=1鈬�2胃=蟺4鈬�胃=蟺8

The area between the curves is,

A=12鈭�伪尾r2d胃A=4脳12鈭�0蟺8sin2胃d胃A=2-cos2胃20蟺8A=2-cos2蟺82+cos202A=1-12units

The required area is A=1-12units

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Most popular questions from this chapter

In Exercises 37鈥�42, find vand find the unit vector in the direction of v.

v=2,1,-5

In Exercises 30鈥�35 compute the indicated quantities when u=(鈭�3,1,鈭�4),v=(2,0,5),andw=(1,3,13)

role="math" localid="1649436488889" (u脳v)路wandu路(v脳w)

Find PQ鈫�

P=1,-2,5,Q=1,-7,-2

Find u+vand u-v. Also sketchu,v,u+vand u-v.

role="math" localid="1649578020551" u=3,-4andv=-1,5

Find PQ鈬赌

P=5,-3,Q=3,-6
See all solutions

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