Question

Let R be the region bounded by the y-axis and the curves y = sin x and y = cos x. Answer the following.
a)Find the exact area of R.
b)A solid is generated by revolving R about the x-axis. Find the exact volume of the solid.

Answer 1

The curves intersect at (pi/4,1/√2)
On that interval, cos(x) > sin(x), so

A = ∫[0,pi/4](cosx-sinx)dx
= sinx+cosx[0,pi/4]
= (1/√2+1/√2)-(0+1) = √2-1

V = ∫[0,pi/4]pi*(R^2-r^2)dx
where R=cosx and r=sinx
V = pi*∫[0,pi/4](cos^2-sin^2)dx
= pi*∫[0,pi/4]cos(2x) dx
= pi/2 sin(2x)[0,pi/4]
= pi/2 * (1-0)
= pi/2