Calculus
posted by Drake on .
R is the region in the plane bounded below by the curve y=x^2 and above by the line y=1.
(a) Set up and evaluate an integral that gives the area of R.
(b) A solid has base R and the crosssections of the solid perpendicular to the yaxis are squares. Find the volume of the solid.
(c) A solid has base R and the crosssections of the solid perpendicular to the yaxis are equilateral triangles. Find the volume of the solid.

(a) The y = x^2 curve crosses y = 1 at x = 1. For the area in question, compute the integral of x^2 from x = 0 to 1, and subtract it from 1.
(b)Integrate along the y axis, with differential slab volume x^2 dy = y dy, from y=0 to y=1
(c) Proceed similary to (b), integrating along y from 0 to 1, but with equilateral triangle slabs. Slab volume must be expressed in terms of x 
Why would it only be the integral of x^2 and not the integral of 1x^2 from x=1 to x=1? So wouldn't that equal an area of 4/3? And secondly, how would you sketch these solids? when you say integrate along the yaxis do you mean to say that the curves have to be shifted as well? And why are all the integrals from x=0 to x=1?

Find the volume, V , of the solid obtained
by rotating the region bounded by the graphs
of
x = y2, x = squareroot(y)
about the line x = −1.