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Use Theorem 9.13 to show that the area of the circle defined by the polar equationr=2acosθisπa2.

Short Answer

Expert verified

The area of the circle isπa2.

Step by step solution

01

Step 1. Given Information

The circle is bounded by the curve r=2acosθ.

02

Step 2. Use the formula for area of a region bounded by a curve 

  • The region is bounded by the curve r=2acosθ.
  • The region is a circle, so the boundary arcs will be θ=0toθ=2π.
  • But the curve is traced twice in this region.
  • So, the area of the region will be calculated as follows:

12×1202π4a2cos2θdθ=a202πcos2θdθ=a202π1+cos2θ2dθ=a2×1202π1dθ+02πcos2θdθ=12a22π+0=πa2

  • Hence, the area of the curve is πa2.

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