Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 14

We use Washer method to evaluate the volume of the solid.

V = int_a^b pi f^2(x) - g^2(x) dx , f(x)gtg(x)
Here
f(x) = sinx, g(x) = cosx

x= 0 and pi/4
about y =-1

V = pi int_0^(pi/4) {[(sinx -(-1)]^2 - [cosx-(-1)]^2}dx

V = pi int_0^(pi/4) {[(sinx + 1]^2 - [cosx + 1]^2}dx

V =pi int_0^(pi/4) [sin^2 x + 2sinx +1 - cosx^2 - 2cosx -1 ]dx

V =pi int_0^(pi/4) [sin^2 x + 2sinx - cos^2 x - 2cosx ]dx

V = pi int_0^(pi/4) [2sinx - 2cosx -(cos^2 x + sin^2 x)]dx

V =pi int_0^ (pi/4) [2sinx - 2cosx - cos2x]dx

V =pi [int_0^(pi/4) 2sinxdx - int_0^(pi/4) 2cosxdx - int_0^(pi/4) cos2xdx]

V = pi [-2cosx - 2sinx - 1/2 sin2x]|_0^(pi/4)


V = pi[-2cos(pi/4) - 2sin(pi/4) - 1/2 sin2* (pi/4) - (-2cos 0 - 2sin0 - 1/2 sin2*0)]
V = pi [-2 * 1/sqrt2 --2 * 1/sqrt2 - 1/2 * 1 - 2*1 - 0 - 0]
V = pi (-2sqrt2 - 3/2)
V = 2sqrt2 pi - 3pi/2
therefore the volume of the solid is 2sqrt2 pi - 3pi/2