41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 3 sin θ and r = 3 cos θ

Verified step by step guidance

1. Identify the curves given in polar coordinates: \(r = 3 \sin \theta\) and \(r = 3 \cos \theta\). These represent two circles in the polar plane.
2. To find the intersection points, set the two equations equal to each other: \(3 \sin \theta = 3 \cos \theta\). Simplify this to \(\sin \theta = \cos \theta\).
3. Solve the equation \(\sin \theta = \cos \theta\) for \(\theta\). This occurs when \(\tan \theta = 1\), so \(\theta = \frac{\pi}{4}\) and also consider the periodicity of tangent to find all relevant solutions within \([0, 2\pi)\).
4. Find the corresponding \(r\) values at the intersection points by substituting \(\theta\) back into either \(r = 3 \sin \theta\) or \(r = 3 \cos \theta\).
5. To find the area of the region inside both curves, set up the integral for the area common to both. The area enclosed by a polar curve \(r(\theta)\) from \(\alpha\) to \(\beta\) is given by \(\frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta\). Determine the limits of integration by analyzing where one curve lies inside the other, then compute the integral of the minimum of the two \(r^2\) values over the appropriate intervals.

Key Concepts

Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle (r, θ), differing from Cartesian coordinates. Understanding how to plot and interpret curves like r = 3 sin θ and r = 3 cos θ is essential for visualizing their shapes and intersection points.

Finding Intersection Points in Polar Coordinates

To find intersections of polar curves, set their equations equal and solve for θ and r. This involves equating r-values and considering the periodic nature of trigonometric functions to identify all points where the curves meet.

Area Calculation Between Polar Curves

The area enclosed by polar curves is found using the integral formula (1/2)∫(r(θ))^2 dθ. For regions bounded by two curves, the area is the integral of the difference of their squared radii over the interval between intersection points.