Area Of Teardrop Shaped Region

Calculus Level 3

The teardrop shaped region in the image is bounded by the curve γ : [ 0 , 2 π ] → R 2 \gamma:[0,2\pi] \to \mathbb{R}^2 with γ ( t ) = ( sin ⁡ ( t ) , ( t − π ) 2 − π 2 ) \gamma(t) = (\sin(t), (t-\pi)^2 - \pi^2) . What is the (unsigned) area of the teardrop shaped region?

Teardrop shaped region given by parametric equations. Teardrop shaped region given by parametric equations.

6 π 6\pi 4 π 4\pi 2 π 2\pi 8 π 8\pi

Sameer Kailasa by Sameer Kailasa , 25, USA

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1 solution

Sb Antu
Jun 5, 2018

The area is integral of x*dy from 0 to 2π x=sin(t) d(y)/dt= 2(t-π) => dy= 2(t-π)dt After integrating 2(t-π)sin(t)dt, the result is -4π. So, the answer here, is 4π.

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