Mathematics

Mathematics

Problem

The curves r=3+2cos⁡θr=3+2\cos\theta and r=4r=4 enclose a region for 0≤θ≤2π0\le\theta\le 2\pi. They meet at θ=π/3\theta=\pi/3 and θ=5π/3\theta=5\pi/3. Which of the following expressions equals the exact area of the region bounded by the two curves? a) 12[∫0π/3 ⁣((3+2cos⁡θ)−4) dθ+∫π/35π/3 ⁣(4−(3+2cos⁡θ)) dθ+∫5π/32π ⁣((3+2cos⁡θ)−4) dθ]\dfrac12\left[\displaystyle\int_{0}^{\pi/3}\!\big((3+2\cos\theta)-4\big)\,d\theta+\int_{\pi/3}^{5\pi/3}\!\big(4-(3+2\cos\theta)\big)\,d\theta+\int_{5\pi/3}^{2\pi}\!\big((3+2\cos\theta)-4\big)\,d\theta\right] b) 12[∫0π/3 ⁣((3+2cos⁡θ)2−42) dθ+∫π/35π/3 ⁣(42−(3+2cos⁡θ)2) dθ+∫5π/32π ⁣((3+2cos⁡θ)2−42) dθ]\dfrac12\left[\displaystyle\int_{0}^{\pi/3}\!\big((3+2\cos\theta)^2-4^2\big)\,d\theta+\int_{\pi/3}^{5\pi/3}\!\big(4^2-(3+2\cos\theta)^2\big)\,d\theta+\int_{5\pi/3}^{2\pi}\!\big((3+2\cos\theta)^2-4^2\big)\,d\theta\right] c) 12[∫0π/3 ⁣(42−(3+2cos⁡θ)2) dθ+∫π/35π/3 ⁣((3+2cos⁡θ)2−42) dθ+∫5π/32π ⁣(42−(3+2cos⁡θ)2) dθ]\dfrac12\left[\displaystyle\int_{0}^{\pi/3}\!\big(4^2-(3+2\cos\theta)^2\big)\,d\theta+\int_{\pi/3}^{5\pi/3}\!\big((3+2\cos\theta)^2-4^2\big)\,d\theta+\int_{5\pi/3}^{2\pi}\!\big(4^2-(3+2\cos\theta)^2\big)\,d\theta\right] d) 12∫02π ⁣((3+2cos⁡θ)2−42) dθ\dfrac12\displaystyle\int_{0}^{2\pi}\!\big((3+2\cos\theta)^2-4^2\big)\,d\theta

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