Volumes by cylindrical shells practice problems
Solids of revolution built from nested cylinders: 2πx f(x) dx. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
The region under \( \displaystyle y = 4 - x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 x^{3} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 x^{2} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \frac{1}{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \sin{\left(x^{2} \right)} \) for \( \displaystyle 0 \le x \le sqrt(pi) \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \frac{3}{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = \frac{1}{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 x^{3} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 \sin{\left(x^{2} \right)} \) for \( \displaystyle 0 \le x \le sqrt(pi) \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 8 - 2 x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 3 x^{2} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
The region under \( \displaystyle y = 2 x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.