Find the area enclosed by the cardioid r = 2(1 + cos θ).
SOLUTION: We graphed the cardioid on our calculator, using polar mode, in
the window [−2,5] × [−3,3] with θ in [0,2π].
Figure N7–8Using the symmetry of the curve with respect to the polar axis we write
B. VOLUME
B1. Solids with Known Cross Sections
If the area of a cross section of a solid is known and can be expressed in terms of
x, then the volume of a typical slice, ΔV, can be determined. The volume of the
solid is obtained, as usual, by letting the number of slices increase indefinitely.
In Figure N7–9, the slices are taken perpendicular to the x-axis so that ΔV = A(x)
Δx, where A(x) is the area of a cross section and Δx is the thickness of the slice.