Question

Find the volume of the solid generated by rotating the region bounded by 𝑦 = 𝑒2𝑥, 𝑥-axis, 𝑦-axis and 𝑥 = ln3 about
(i) the 𝑥-axis for 1 complete revolution.
(ii) the 𝑦-axis for 1 complete revolution.
(iii) 𝑦 = −1 for 1 complete revolution.
(iv) 𝑥 = −1 for 1 complete revolution.

Answer 1

as usual, the volumes can be done using shells (v = 2πrh) or discs (washers) (v=πr^2), both multiplied by the thickness (dx or dy). I'll show both integrals

(i)
v = ∫[0,ln3] π(e^(2x))^2 dx = 20π
v = π(1^2)(ln3) + ∫[1,9] 2πy(ln3 - 1/2 lny) dy = ln3 π + 20π - ln3 π = 20π

(ii)
v = π(ln3)^2 + ∫[1,9] π((ln3)^2 - (1/2 lny)^2) dy = (9log3-4)π
v = ∫[0,ln3] 2πxe^(2x) dx = (9log3-4)π

Do the others the same way, but the radius is increased by 1.