A solid has a circular base of radius 4cm. Find the volume of the solid if every plane section perpendicular to the x-axis is a square. I'm having trouble setting the integral up. 0 to 8 integral pir^2 dx??
[Math] Volume using cross sections 21
calculusdefinite integralsvolume
Related Solutions
The cross sections perpendicular to the $x$-axis start at $x=0$ and end at $x=1$. Let $A(x)$ denote the area of the cross section perpendicular to the $x$-axis that intersects the $x$-axis at $x$. The solid can be thought of as the "sum of the cross sections" as they range from $x=0$ to $x=1$. The volume of the solid is thus $$\int_{0}^1 A(x)\,dx.$$
We need to find an explicit expression for $A(x)$ in terms of $x$. We know that the cross-section at $x$ is a square. The bottom vertices of this square are bounded by the parabola $x=1-y^2$. So, the side length $\color{maroon}{\ell_x}$ of the square is $\color{maroon}{\ell_x}=2\sqrt{1-x}$; and thus, $$A(x)=(\,2\sqrt{1-x}\,)^2.$$ So the volume is $$\int_{0}^1 (\,2\sqrt{1-x}\,)^2 \,dx = \int_{0}^1 4(1-x) \,dx .$$
In the diagram below, we are staring down the $z$-axis. The base of the solid is enclosed by the green curve. A cross section at $\color{maroon} x$ would appear as the line segment $\color{maroon}{\ell_x}$. This cross section is a square with side length $\color{maroon}{\ell_x}=2\color{maroon}{\sqrt{1-x}}$.
Since this has gotten bumped, here's a diagram for posterity, with a thousand words omitted.
Best Answer
We will be integrating with respect to $x$, from the leftmost value of $x$, which is $-4$, to the rightmost value, which is $4$. The volume is $$\int_{x=-4}^4 A(x)\,dx,$$ where $A(x)$ is the area of cross-section at $x$.
The cross-section is a square. Let us find its side. Note that the circle has equation $x^2+y^2=16$.
At $x$, the cross-section extends from $y=-\sqrt{16-x^2}$ to $y=\sqrt{16-x^2}$. So the square has side $2\sqrt{16-x^2}$, and therefore $A(x)=4(16-x^2)$.
The rest is straightforward integration. I would prefer to use symmetry, and instead of integrating from $-4$ to $4$, integrate from $0$ to $4$ and double the answer.
Remark: To begin, we need to visualize the solid. The base is circular. Imagine the thing sitting on a table. The bottom is a circle. The solid extends upwards, and is "tall" near $x=0$, and tapers off to height near $0$ when $x$ gets close to $-4$ and $4$. Kind of like a hill, except that there are vertical cliffs on the north and south side.