Show that the volume of the solid formed by the curve y=cos(x/2), as it is rotated 360° around the x-axis between x= π/4 and x=3π/4, is of the form π^2/a. Find the constant a.
After sketching a diagram of the curve and the solid for clarity, we see that we need to use the formula V = π∫ y2 dx (with upper and lower bounds of 3π/4 and π/4 respectively) to calculate the volume of the solid formed by the revolution. If we replace y with its expression in terms of x, we obtain integral π∫ (cos(x/2)2 dx so V = π∫ cos2(x/2) dx. Using the double angle formula (cos(2θ) = 2cos2(θ)-1), we get cos2(θ) = cos(2θ)/2+1/2 by rearranging. And we can set θ = x/2 to give us cos2(x/2) = cos(x)/2+1/2, which can be substituted into our integral and be evaluated as follows:V = π∫ (cos(x)/2+1/2) dx = π[sin(x)/2+x/2] (with same bounds as earlier)= π[(sin(3π/4)/2+3π/8)-(sin(π/4)/2+π/8) (after evaluating at boundaries)= π(√2/4+3π/8-√2/4-π/8) = π(π/4) = π2/4Which is in the required form, so a=4.
