Volumes with known cross sections practice problems
Volumes of solids whose cross-sections are squares, semicircles or triangles. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
The base of a solid is the disk \( \displaystyle x^2 + y^2 \le 4 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 9 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are semicircles. Find the volume of the solid.
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 16 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.
The base of a solid is the region between \( \displaystyle y = x \) and \( \displaystyle y = x^2 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (3,0) \) and \( \displaystyle (0,6) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are semicircles. Find the volume of the solid.
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (2,0) \) and \( \displaystyle (0,5) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.
The base of a solid is the disk \( \displaystyle x^2 + y^2 \le 16 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
The base of a solid is the region between \( \displaystyle y = 4 - x^{2} \) and the \( \displaystyle x \)-axis. Cross-sections perpendicular to the \( \displaystyle x \)-axis are semicircles. Find the volume of the solid.
The base of a solid is the region between \( \displaystyle y = 1 - x^{2} \) and the \( \displaystyle x \)-axis. Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.
The base of a solid is the region between \( \displaystyle y = 4 - x^{2} \) and the \( \displaystyle x \)-axis. Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.