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Prove that the area of a circle of radius risπr2, as follows:

(a) Write down a definite integral that represents the area of the circle of radius rcentered at the origin. (Hint: The equation of such a circle is x2+y2=r2.)

(b) Use trigonometric substitution to solve the definite integral you found in part (a). (Hint: Change the limits of integration to match your substitution .)

Short Answer

Expert verified

Part (a) The integral is 2-rrr2-x2dx.

Part (b) The value isπr2.

Step by step solution

01

Part (a) Step 1. Given information.

The equation of the circle isx2+y2=r2.

02

Part (a) Step 2. Explanation.

The definite integral will be,

x2+y2=r2y2=r2-x2y=r2-x22-rrr2-x2dx

03

Part (b) Step 1. Explanation.

On solving,

2-rrr2-x2dx=2-π2π2r2-r2sin2u(rcosu)du=2r2-π2π2cos2udu=2r212-π2π2(1+cos2u)du=r2u+12sin(2u)-π2π2=πr2

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