Find the volume of the described solid.
-The base of the solid is the disk $x ^ { 2 } + y ^ { 2 } \leq 4$. The cross sections by planes perpendicular to the $y$-axis between $y = - 2$ and $y = 2$ are isosceles right triangles with one leg in the disk.
A) $\frac { 64 } { 3 }$
B) $\frac { 16 } { 3 }$
C) $\frac { 80 } { 3 }$
D) $\frac { 128 } { 3 }$

Question

Quizplus · Accepted Answer

The volume of the solid can be calculated using the method of cross-sections. The area of an isosceles right triangle with leg length $l$ is $\frac{1}{2}l^2$. For a given value of $y$, the length of the leg of the triangle is the diameter of the slice of the disk at that $y$, which is $2\sqrt{4-y^2}$ because the equation of the disk is $x^2 + y^2 = 4$. Thus, the area of a cross-section is $\frac{1}{2}(2\sqrt{4-y^2})^2 = 2(4-y^2)$. The volume is the integral of the cross-sectional area from $y = -2$ to $y = 2$, which is $\int_{-2}^{2} 2(4-y^2) dy = \left[ 8y - \frac{2}{3}y^3 \right]_{-2}^{2} = \left(16 - \frac{16}{3}\right) - \left(-16 + \frac{16}{3}\right) = \frac{64}{3}$.

Find the Volume of the Described Solid x2+y2≤4x ^ { 2 } + y ^ { 2 } \leq 4x2+y2≤4

Find the Volume of the Described Solid $x ^ { 2 } + y ^ { 2 } \leq 4$