Find an integral expression for the area of the shaded region.
We can see from the picture that the limits of integration go from to .
We slice the region into tiny slices and pretend the slice at angle θ is a slice of a perfect circle with radius
r = sinθ + 1
for that particular value of θ. The area of the entire perfect circle is
π(sinθ + 1)2
and the area of the tiny slice is a fraction of that:
Adding up the areas of the tiny slices and letting the number of slices approach infinity, we see that the area of the shaded region is