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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area enclosed by the curve in Exercise 29.

To determine

To find: The area of the region which is enclosed by the curve x=2acostacos2t and y=2asintasin2t .

Explanation

Given:

The Cartesian equation for the variable x is as follows.

x=2acostacos2t (1)

The Cartesian equation for the variable y is as follows.

y=2asintasin2t (2)

Calculation:

Area is determined between the limits of 0 to 2π .

Calculate the area enclosed inside the loop by using the formula.

A=abydx (3)

Differentiate equation (2) with respect to t .

dxdt=2asint(2asin2t)dxdt=2asint+2asin2t

dx=(2asint+2asin2t)dt (4)

Substitute the expressions from equation (4) and (1) in equation (3).

A=02π(2asintasin2t)(2asint+2asin2t)dt=02π(4a2sin2t+2a2sintsin2t+4a2sintsin2t2a2sin22t)dt=02π(4a2sin2t+6a2sintsin2t2a2sin22t)dt=02π(4a2sin2t+6a2sintsin2t2a2sin22t)dt=02π(4a2sin2t6a2sintsin2t+2a2sin22t)dt

A=02π(4a2sin2t)dt02π(6a2sintsin2t)dt+02π(2a2sin22t)dt (5)

Integrate each term separately as below.

02π4a2sin2tdt=4a2sin2tdt=4a21+cos2t2dt=2a21+cos2tdt=2a2[t12sin(2t)]02π=[2a2t+a2sin(2t)]02π=[(2a2(2π)+a2sin(2(2π)))(2a2(0)+a2sin(2(0)))]

02π4a2sin2

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