Let 𝑆 be the region (in the first quadrant) bounded by a circle 𝑥^2 + 𝑦^2 = 2, 𝑦^2 = 𝑥 and the 𝑥-axis

Question

(ii) Find the volume of the solid generated by rotating the region 𝑆 about the 𝑦-axis
(c) Find the surface area of the solid generated by rotating the region 𝑆 about the 𝑥-axis.



Answer 1

(i)
v = ∫[0,1] π(R^2-r^2) dy
where R^2=2-y^2 and r=y^2

(ii)
v = ∫[0,1] 2πrh dy
where r=y and h=√(2-y^2)-y^2