area by integration

Problem
A circle of radius r rolls along a horizontal line without skidding.

1. Find the equation traced by a point on the circumference of the circle.
2. Determine the length of one arc of the curve.
3. Calculate the area bounded by one arc of the curve and the horizontal line.

Problem
A goat is tied outside a silo of radius 10 m by a rope just long enough for the goat to reach the opposite side of the silo. Find the area available for grazing by the goat. Note that the goat may not enter the silo.
A. 1033.54 m²
B. 2067.08 m²
C. 516.77 m²
D. 2583.86 m²
 

Problem
Find the area bounded by $r = a \sin 3\theta$ and $r = a \cos 3\theta$.
 

Problem
Find the area enclosed by the following:

Problem
Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.
 

Example 6
What is the area within the curve r² = 16 cos θ?
 

Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).
 

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
 

Problem
Find the area individually enclosed by the following Cardioids:
(A)   $r = a(1 - \cos \theta)$
(B)   $r = a(1 + \cos \theta)$
(C)   $r = a(1 - \sin \theta)$
(D)   $r = a(1 + \sin \theta)$
 

Example 2
Find the area bounded by the lemniscate of Bernoulli r² = a² cos 2θ.