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1. **8.33:** The area \( A \) of a region is the integral of the square of the distance of each point in the region from the origin. Compute \( A \) when the region is the portion of the plane between the circles \( r = 1 \) and \( r = 2 \).

2. **8.34:** The area of the region bounded by the cardioid \( r = 1 + \cos(\theta) \) can be written as 
   \[
   A = \int_a^b f(\theta) \, d\theta.
   \]
   Set up, but do not evaluate, the integral that represents the area of the region enclosed by this cardioid.

3. **8.35:** Compute the arc length of the curve given in polar form by the equation \( r = 2 + 3\sin(\theta) \) for \( 0 \leq \theta \leq \pi \).

4. **8.36:** Find the volume of the solid obtained by

The original question submitted by the user (calculus-related problems involving regions, polar coordinates, integrals).

This page provides solutions to calculus problems involving areas, arc lengths, and volumes using polar coordinates and integrals. Topics covered include the cardioid, regions between circles, and arc length of polar curves.

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