Answer to Question #98857 in Calculus for Rachel

Question #98857
Use Green’s Theorem to find the area inside x^2/3+y^2/3= 1.
1
Expert's answer
2019-11-21T10:53:57-0500

Green's Theorem

We need to find the Area inside x23+y23=1{x^\frac {2} {3}} + {y^\frac {2} {3}} = 1


Solution:


Given,

x23+y23=1{x^\frac {2} {3}} + {y^\frac {2} {3}} = 1

Let

x=cos3 t   and y=sin3 t, 0t2πx = cos^{ 3} \space {t} \space \space \space and \space y = sin^ {3} \space {t} , \space 0\le t \le 2\pi

By Green's theorem


Area inside =

12A(x dyy dx)\frac {1}{2} \oint_A ( x \space dy - y \space dx)

We know,

x=cos3 t x = cos^{3} \space {t} \space

dx=3 cos2 (t)×(sin(t))=3cos2(t)sin(t) dtdx = {3} \space cos^{2} \space {(t)} \times (-sin (t)) = - 3 cos^2 (t) sin (t) \space dt




y=sin3 ty = sin^ {3} \space {t}


dy=3sin2(t)×cos(t) dtdy = 3 sin^2 (t) \times cos (t) \space dt



Now plug all these in the formula of Green's theorem


 Area inside  = 12A((x dyy dx)=12[t=02π(cos3 t3sin2(t)×cos(t) dt)sin3 t(3cos2(t)sin(t)) dt)]\frac {1}{2} \oint_A (( x \space dy - y \space dx) = \frac {1}{2} [\int_{t =0} ^ {2\pi} (cos^{ 3} \space {t} 3 sin^2 (t) \times cos (t) \space dt) - sin^ {3} \space {t} (- 3 cos^2 (t) sin (t)) \space dt)]


=32[t=02π(cos4 t sin2(t)× dt)+sin4 t cos2(t) dt)]= \frac {3}{2} [\int_{t =0} ^ {2\pi} (cos^{ 4} \space {t} \space sin^2 (t) \times \space dt) + sin^ {4} \space {t} \space cos^2 (t) \space dt)]


=32[t=02πcos2 t sin2(t)× (cos2t+sin2 t)]dt= \frac {3}{2} [\int_{t =0} ^ {2\pi} cos^{ 2} \space {t} \space sin^2 (t) \times \space (cos^2 t + sin^ {2} \space {t} ) ] dt

=32t=02πcos2 t sin2(t)dt= \frac {3}{2} \int_{t =0} ^ {2\pi} cos^{ 2} \space {t} \space sin^2 (t) dt

=32×2×2t=0π2cos2 t sin2(t)dt= \frac {3}{2} \times 2 \times 2 \int_{t =0} ^ {\frac {\pi} {2}} cos^{ 2} \space {t} \space sin^2 (t) dt

=6t=0π2cos2 t sin2(t)dt= 6 \int_{t =0} ^ {\frac {\pi} {2}} cos^{ 2} \space {t} \space sin^2 (t) dt

=64t=0π2(2sintcost)2dt==64t=0π2sin2 2t dt= \frac {6} {4} \int_{t =0} ^ {\frac {\pi} {2}} ( 2 sin t cos t)^2 dt = = \frac {6} {4} \int_{t =0} ^ {\frac {\pi} {2}} sin ^2 \space 2t \space dt

( Since sin 2t = 2 sint t cos t)


=64t=0π21cos4t2 dt=34[tsin4t4]t=0t=π2= \frac {6} {4} \int_{t =0} ^ {\frac {\pi} {2}} \frac { 1- cos 4t} {2} \space dt = \frac {3}{4} [ t - \frac {sin 4t}{4}]_{t =0} ^{ t = \frac {\pi}{2}}

=34×π2=3π8= \frac {3}{4} \times \frac {\pi}{2} = \frac {3 \pi} {8}

Answer: The Area inside = 3π8\frac {3 \pi} {8}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment