Volumes with Cross Sections (College Board AP® Calculus AB): Exam Questions

Let $R$ be the region bounded below by the graph of $y=\cos x$ and above by the graph of $y=\sin x$, between $x=\frac{\pi }{4}$ and $x=\frac{5\pi }{4}$. $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?

The base of a solid is a region in the first quadrant bounded by the $x$-axis, the $y$-axis, and the line $3x+5y=15$, as shown in the figure above. If cross sections of the solid perpendicular to the $x$-axis are semicircles, what is the volume of the solid?

The base of a solid is the region in the first and fourth quadrants bounded by the lines $y=x$ and $y=-x$ between $x=0$ and $x=5$. If cross sections of the solid perpendicular to the $x$-axis are rectangles, with the height of each rectangle equal to one half of its base, which of the following integrals would correctly calculate the volume of the solid?

The base of a solid is the region in the first quadrant bounded by the $y$-axis, the graph of $y=\arctan \left(2x\right)$, the horizontal line $y=4$, and the vertical line $x=1.5$. For this solid, each cross-section perpendicular to the $x$-axis is a square. What is the volume of the solid?

Let $R$ be the shaded region in the first quadrant bounded by the graphs of $y=2\cos \left(\frac{\pi x}{2}\right)$ and $y={\left(x-\frac{3}{2}\right)}^2-\frac{1}{4}$, as shown in the figure below. The region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in region $R$. What is the volume of the solid?

The base of a solid is the region in the first quadrant bounded by the curves $y=x^2$ and $y=\sqrt{x}$. If cross sections of the solid perpendicular to the $x$-axis are triangles, with the height of each triangle equal to its $x$-coordinate, which of the following integrals would correctly calculate the volume of the solid?

The base of a solid is the region enclosed by the circle given by $x^2+y^2=r^2$. If cross sections of the solid perpendicular to the $x$-axis are squares, calculate the volume of the solid in terms of $r$.

Let $R$ be the shaded region in the first quadrant bounded by the graphs of $y=f\left(x\right)$, $y=g\left(x\right)$, and the line $x=1$, as shown in the diagram below.

The functions $f$ and $g$ are defined as $f\left(x\right)=4e^x$ and $g\left(x\right)=4e^{-x}$.

The region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with the hypotenuse in region $R$. Which integral describes the volume of the solid?