Question

Let R be the triangular region in the first quadrant, with vertices at points (0,0), (0,2), and (1,2). The region R is the base of a solid. For the solid, each cross section perpendicular to the y-axis is an isosceles right triangle with the right angle on the y-axis and one leg in the xy-plane. What is the volume of the solid?13

Answer 1

The line through (0, 0) and (1, 2) has equation $y=2x$.

Each cross section has a leg in the $xy$-plane whose length is the horizontal distance from the $y$-axis to the line $y=2x$, which is $x=\frac y2$.

The cross sections are isosceles right triangles, so the legs that lie perpendicular to the $xy$-plane have the same length as the legs *in* the $xy$-plane, hence these triangles haves bases and heights equal to $\frac y2$. Then the area of each cross section is

$\dfrac12\left(\dfrac y2\right)^2=\dfrac{y^2}8$

where the cross sections are generated over the interval $0\le y\le2$.

The volume of the solid is then

$\displaystyle\int_0^2\frac{y^2}8\,\mathrm dy=\frac1{24}y^3\bigg|_0^2=\frac{2^3}{24}=\frac13$