A heartful of problem

Calculus Level 4

Find area occupied by the "circle" whose radius is given by r = 1 + sin ⁡ θ r= 1+\sin \theta .

Give your answer to 3 decimal places.


The answer is 4.71238898038.

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A Former Brilliant Member by A Former Brilliant Member

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1 solution

∫ 0 2 π 1 2 ( 1 + sin ⁡ θ ) 2 d x \int _{ 0 }^{ 2\pi }{ \frac { 1 }{ 2 } \left( 1+\sin { \theta } \right) ^{ 2 } dx} = 3 ( π ) / 2 = 3 (\pi ) /2

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Why ?? I cant understand ??

Kushal Bose - 4 years, 9 months ago

To put it simple, let's consider what we do in integration by its definition, we use the Rectangle method, something that divides area under curve into infinitesimal slices. The problem now is to find the area of cardioid, the heart-shaped circle. The matter now is to divide the area again to infinite slices, with each characterised by their "radius". Area of sector can be given by 1 / 2 ( R a d i u s ) 2 ( θ ) 1/2(Radius)^2(\theta ) . Using Riemann sums, we obtain ∫ 0 2 π 1 2 ( 1 + sin ⁡ θ ) 2 d x \int _{ 0 }^{ 2\pi }{ \frac { 1 }{ 2 } \left( 1+\sin { \theta } \right) ^{ 2 } dx} = 3 ( π ) / 2 = 3 (\pi ) /2 .

A Former Brilliant Member - 4 years, 9 months ago

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