33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

Verified step by step guidance

1. First, recognize that the given curve is in polar coordinates: \(r = 8 \sin \theta\). This represents a circle in the polar plane.
2. Sketch the curve by plotting points for various values of \(\theta\) between \$0\( and \)2\pi\(. Note that \)r\( is non-negative when \)\sin \theta \geq 0\(, which occurs between \)0\( and \)\pi$.
3. Recall that the area enclosed by a polar curve \(r = f(\theta)\) from \(\theta = a\) to \(\theta = b\) is given by the formula: \[\text{Area} = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta\]
4. Substitute \(r = 8 \sin \theta\) into the formula to get: \[\text{Area} = \frac{1}{2} \int_{0}^{\pi} (8 \sin \theta)^2 \, d\theta = \frac{1}{2} \int_{0}^{\pi} 64 \sin^2 \theta \, d\theta\]
5. Simplify the integral and use the identity \(\sin^2 \theta = \frac{1 - \cos 2\theta}{2}\) to evaluate the integral over \([0, \pi]\). This will give the area of the region inside the circle.

Key Concepts

Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle (r, θ). Understanding how to plot the curve r = 8 sin θ helps visualize the region bounded by the circle, which is essential for setting up the integral for the area.

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by (1/2) ∫[a to b] (r(θ))^2 dθ. This formula is crucial for finding the area inside the circle r = 8 sin θ by integrating over the appropriate interval.

Determining the Limits of Integration

Identifying the correct interval for θ where the curve traces the region is necessary to compute the area accurately. For r = 8 sin θ, the curve is traced as θ goes from 0 to π, which defines the bounds for the integral.