Mathematics

Mathematics

Problem

For the region bounded by the outer curve r=f(θ)r=f(\theta) and the inner curve r=g(θ)r=g(\theta) on the interval α≤θ≤β\alpha\le \theta\le\beta with f(θ)≥g(θ)≥0f(\theta)\ge g(\theta)\ge0, which integral gives the exact area of the region? a) ∫αβ ⁣(f(θ)2−g(θ)2) dθ\displaystyle \int_{\alpha}^{\beta}\!\bigl(f(\theta)^2-g(\theta)^2\bigr)\,d\theta b) 12∫αβ ⁣(f(θ)−g(θ))2 dθ\displaystyle \tfrac12\int_{\alpha}^{\beta}\!\bigl(f(\theta)-g(\theta)\bigr)^2\,d\theta c) 12∫αβ ⁣(f(θ)2−g(θ)2) dθ\displaystyle \tfrac12\int_{\alpha}^{\beta}\!\bigl(f(\theta)^2-g(\theta)^2\bigr)\,d\theta d) 12∫αβ ⁣f(θ)2 dθ\displaystyle \tfrac12\int_{\alpha}^{\beta}\!f(\theta)^2\,d\theta

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