26. which of the following expressions gives the total area enclosed by the polar curve r = sin²θ shown in…

26. which of the following expressions gives the total area enclosed by the polar curve r = sin²θ shown in the figure above? (a) 1/2∫₀^π sin²θ dθ (b) ∫₀^π sin²θ dθ (c) 1/2∫₀^π sin⁴θ dθ (d) ∫₀^π sin⁴θ dθ (e) 2∫₀^π sin⁴θ dθ

26. which of the following expressions gives the total area enclosed by the polar curve r = sin²θ shown in the figure above? (a) 1/2∫₀^π sin²θ dθ (b) ∫₀^π sin²θ dθ (c) 1/2∫₀^π sin⁴θ dθ (d) ∫₀^π sin⁴θ dθ (e) 2∫₀^π sin⁴θ dθ

Answer

Explanation:

Step1: Recall area formula in polar coordinates

The formula for the area $A$ enclosed by a polar - curve $r = f(\theta)$ from $\theta=\alpha$ to $\theta = \beta$ is $A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta$.

Step2: Identify $r$ and limits of integration

Given $r = \sin^{2}\theta$, and for a complete loop of the polar curve, we can take the limits of integration from $\theta = 0$ to $\theta=\pi$. Substitute $r=\sin^{2}\theta$ into the area formula: $A=\frac{1}{2}\int_{0}^{\pi}(\sin^{2}\theta)^{2}d\theta=\frac{1}{2}\int_{0}^{\pi}\sin^{4}\theta d\theta$.

Answer:

C. $\frac{1}{2}\int_{0}^{\pi}\sin^{4}\theta d\theta$