Question

1.multiple - choice(5 points) medium what is the formula for the area a enclosed by a polar curve r = φ(θ) between the angles θ = α and θ = β? a | a = ∫_{α}^{β} r dθ b | a = 1/2 ∫_{α}^{β}φ(θ)² dθ c | a = ∫_{α}^{β} φ(θ) dθ d | a = 1/2 ∫_{α}^{β} r dθ

Explanation:

Step1: Recall polar - area formula

The formula for the area $A$ of the region bounded by the polar curve $r = \varphi(\theta)$ and the rays $\theta=\alpha$ and $\theta = \beta$ is derived from the formula for the area of a sector of a circle $A=\frac{1}{2}r^{2}\Delta\theta$. By using the definite - integral to sum up the areas of infinitesimal sectors, we get $A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta$. Since $r = \varphi(\theta)$, the formula is $A=\frac{1}{2}\int_{\alpha}^{\beta}[\varphi(\theta)]^{2}d\theta$.

Answer:

B. $A=\frac{1}{2}\int_{\alpha}^{\beta}[\varphi(\theta)]^{2}d\theta$