Compute-the-area-of-a-loop-of-the-curve-asin-2-and-even-sketch-the-curve-please-

Question Number 23283 by ajfour last updated on 28/Oct/17

C o m p u t e t h e a r e a o f a l o o p o f

t h e c u r v e ρ = a \sin 2 θ; a n d e v e n

s k e t c h t h e c u r v e, p l e a s e .

Answered by mrW1 last updated on 28/Oct/17

a small loop for θ from 0 to \frac{π}{2} :

A_{1} = \frac{1}{2} \int_{0}^{\frac{π}{2}} ρ^{2} d θ

= \frac{a^{2}}{2} \int_{0}^{\frac{π}{2}} \sin^{2} 2 θ d θ

= \frac{a^{2}}{4} \int_{0}^{\frac{π}{2}} (1 - \cos 4 θ) d θ

= \frac{a^{2}}{4} {[θ - \frac{\sin 4 θ}{4}]}_{0}^{\frac{π}{2}}

= \frac{π a^{2}}{8}

for a total loop from 0 to 2 π,

A = 4 \times \frac{π a^{2}}{8} = \frac{π a^{2}}{2}

Commented by mrW1 last updated on 28/Oct/17

Commented by ajfour last updated on 28/Oct/17

t h a n k y o u v e r y m u c h S i r, r i g h t

a n s w e r, n i c e q u e s t i o n; i t h o u g h t

o f s h a r i n g i t w i t h y o u .

i^{'} v e c r e a t e d a c a r o m b o a r d q u e s t i o n,

p l e a s e v i e w i t, s i r .

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