Mathematics

Mathematics

Problem

In polar coordinates, let two curves be defined by r=f(θ)r=f(\theta) and r=g(θ)r=g(\theta) for α≤θ≤β\alpha\le\theta\le\beta, with f(θ)≥g(θ)f(\theta)\ge g(\theta) on that interval. Which of the following expressions gives the area of the region between the two curves? 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)2−g(θ)2] dθ\displaystyle \tfrac12\int_{\alpha}^{\beta}\bigl[f(\theta)^2-g(\theta)^2\bigr]\,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)−g(θ)]2 dθ\displaystyle \tfrac12\int_{\alpha}^{\beta}\bigl[f(\theta)-g(\theta)\bigr]^2\,d\theta

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