Volumes by disks and washers practice problems
Solids of revolution sliced perpendicular to the axis. 18 problems with worked solutions; in 18 of them every equation is proved by a computer algebra system.
The region under \( \displaystyle y = \frac{1}{x} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \) is revolved about the x-axis. Find the volume.
The region between \( \displaystyle y = x \) and \( \displaystyle y = x^{2} \) is revolved about the x-axis. Find the volume.
The region between \( \displaystyle y = 2 x \) and \( \displaystyle y = x^{2} \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \sqrt{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \frac{1}{x} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x^{2} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \sin{\left(x \right)} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = \pi \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \sqrt{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = e^{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x^{2} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 4 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x^{2} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = e^{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \frac{1}{x} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = x \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = \sqrt{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 1 \) is revolved about the x-axis. Find the volume.
The region under \( \displaystyle y = e^{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.