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Chapter 6: Q7P (page 314)

Use Problem 6 to show that the area inside the ellipse x=acos胃,y=bsin胃,0鈮�胃鈮�2蟺,isA=蟺ab.

Short Answer

Expert verified

The solution to this problem isA=ab蟺..

Step by step solution

01

Given Information.

The given information is as follows:

x=acos胃,y=bsin胃.

02

Definition of Green’s Theorem.

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

03

Find the solution.

Write the given conditions.

x=acos胃,y=bsin胃.

Write the differentiation.

dx=-asin胃d胃dy=bcos胃d胃

Find the area.

A=12鈭�02蟺abcos2胃+sin2胃d胃=2蟺ab2=蟺ab

Hence, the solution to this problem is role="math" localid="1659154628756" A=ab蟺

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Most popular questions from this chapter

鈭�F.draround the circle over the curved part of the hemisphere in Problem 24, if x2+y2+2x=0, where F=yi-xj.

Let F = i - 5j + 2kact at the point (2, 1, 0) Find the torque of F about the line -(3j + 4k)- 2it .

Find the derivative of zexcosy at (1,0,蟺3) in the direction of the vector i+2j .

Question: 鈭�V路draround the circle(x-2)2+(y-3)2=9,z=0whereV=(x2+yz2)i+(2x-y3)j

鈭�F脳苍诲蟽where F=(y2鈭�x2)i+(2xy鈭�y)j+3zkand 蟽is the entire surface of the tin can bounded by the cylinder

role="math" localid="1657353627256" x2+y2=16

role="math" localid="1657353639412" z=3

role="math" localid="1657353647648" z=-3
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