Question

Given a parametric curve

{x = 2 cost
{y = 4 sint 0 <= t <= π

a. Set up but do NOT evaluate an integral to find the area of the region enclosed by the curve and the x-axis.
b. Set up but do NOT evaluate an integral to find the area of the surface obtained by rotating the curve about the x-axis.

Answer 1

(a) The area of the region would be given by the integral

`\displaystyle\int_0^\pi y(t)\left|x'(t)\right|\,\mathrm dt = 8 \int_0^\pi \sin^2(t)\,\mathrm dt`

(b) The area of the surface of revolution would be given by

`\displaystyle\int_0^\pi y(t)√(x'(t)^2+y'(t)^2)\,\mathrm dt = 4\int_0^\pi\sin(t)√(4\sin^2(t)+16\cos^2(t))\,\mathrm dt`