Let region R be the shaded region in the first quadrant enclosed by the graphs of y=sinx y=cosx, and the y axis

Question

find the area of region R, find the volume of the solid generated when R is revolved around the x and y axis

Answer 1

Area s = ∫₀^π/4sinxdx + ∫_π/4^π/2cosxdx = - cosx₀^π/4 + sinx_π/4^π/2 = 1 - √2/2 + 1 - √2/2 = 2 - √2

Volume about x-axis: v_x = π(∫₀^π/4sin²xdx + ∫_π/4^π/2cos²xdx) = π[(1/2x - 1/4sin2x)₀^π/4 + (1/2x + 1/4sin2x)_π/4^π/2] = π[(π/8 - 1/4) + π/4 - π/8 - 1/4] = π(π/4 - 1/2) = π(π - 2)/4

Volume about y-axis: v_y = 2π[∫₀^π/4xsinxdx + ∫_π/4^π/2xcosxdx] = 2π[(-xcosx + sinx)₀^π/4 + (xsinx + cosx)_π/4^π/2] =

2π[(- π√2/8 + √2/2) + π/2 - π√2/8 - √2/2] = π²(2 - √2)/2