Calculus I, Fall 2021

Brief solutions to Quiz 7

Dec 21, 2021

1. Find the volume of the region bounded by z ≥ 0, z ≤ x and x^{2} + y^{2} ≤ 9 (i.e. above the xy plane, below the plane x = z and inside the cylinder x^{2} + y^{2} = 9). Ans: See page 2-3 of Lecture 21 note.

2. Use two different methods (disks/washers and cylindrical shells) to find the volume of revolution obtained by rotating the region {0 ≤ x ≤ 2, x^{2} ≤ y ≤ 2x} around the xaxis. Ans: Method of disks(washers):

V = ∫_{0}^{2} π((2x)^{2} − (x^{2})^{2})dx = π(\frac{4}{3}x^{3} − \frac{1}{5}x^{5})|_{0}^{2} = \frac{64}{15}π

Method of cylindrical shells:

V = ∫_{0}^{4} 2πy(\sqrt{y} − \frac{y}{2})dy = 2π(\frac{2}{5}y^{\frac{5}{2}} − \frac{1}{6}y^{3})|_{0}^{4} = \frac{64}{15}π

3. Same as above, but change the rotation axis to y- axis.

Ans: Method of disks(washers):

V = ∫_{0}^{4} π((\sqrt{y})^{2} − (\frac{y}{2})^{2})dy = π(\frac{1}{2}y^{2} − \frac{1}{12}y^{3})|_{0}^{4} = \frac{8}{3}π

Method of cylindrical shells:

V = ∫_{0}^{2} 2πx(2x − x^{2})dx = 2π(\frac{2}{3}x^{3} − \frac{1}{4}x^{4})|_{0}^{2} = \frac{8}{3}π