NEW! Section 6.2 – Volumes (Disks, Washers, and by Slicing)
Instructions
- A video explaining the concepts for this section is coming soon.
- This page includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
- You can find additional practice problems for this section on the Practice Problems page for this Module
Concepts
- Finding the area between two curves on a closed interval
- Finding the area bounded by two curves
- Determining whether to integrate along the \(x\) or \(y\)-axis when finding the area between curves
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Sketch the region \(R\) bounded by \(y=e^x\), \(x=-1\), \(x=2\), \(y=0\). Find the volume of the solid obtained by rotating \(R\) about the \(x\) axis.
\(\displaystyle \frac{\pi}{2}\left[e^4-e^{-2}\right]\)
See the video for a sketch of the region.
See the video for a sketch of the region.
2. Sketch the region \(R\) bounded by \(y=x^3\), \(y=8\), \(x=0\). Find the volume of the solid obtained by rotating \(R\) about the \(y\) axis, then about the line \(y=8.\)
Rotating about the \(y\) axis, the volume is \(\displaystyle \frac{96\pi}{5}.\)
Rotating about the line \(y=8\), the volume is\(\displaystyle \frac{576 \pi}{7}.\)
See the video for a sketch.
Rotating about the line \(y=8\), the volume is\(\displaystyle \frac{576 \pi}{7}.\)
See the video for a sketch.
3. Sketch the region \(R\) bounded by \(y=\sqrt{x}\), \(y=0\), \(x=2\). Find the volume of the solid obtained by rotating \(R\) about the line \(x=2.\)
The volume is \(\displaystyle \pi\left[ \frac{32\sqrt{2}}{15}\right].\)
See the video for a sketch.
See the video for a sketch.
4. Consider the region \(R\) bounded by \(y=x^2\), \(y=2x\). Set up but do not evaluate an integral that gives the volume obtained by rotating \(R\) about the line \(x=3.\)
\(V=\displaystyle \int_0^4 \pi \left[ \left( 3-\frac{y}{2}\right)^2-\left(3-\sqrt{y}\right)^2\right]\, dy\)
5. Consider the region \(R\) bounded by \(y=x^2\), \(y=2x\). Set up but do not evaluate an integral that gives the volume obtained by rotating \(R\) about the line \(x=-2.\)
\(\displaystyle V=\int_0^4 \pi \left[ \left( \sqrt{y}+2\right)^2-\left( \frac{y}{2}+2\right)^2\right] \, dy\)
6. Consider the region \(R\) bounded by \(y=x^2\), \(y=2x\). Set up but do not evaluate an integral that gives the volume obtained by rotating \(R\) about
- The line \(y=5.\)
- The line \(y=-1.\)
- \(V=\displaystyle \int_0^2 \pi\left[ \left(5-x^2\right)^2-\left(5-2x\right)^2\right]\, dx\)
- \(V=\displaystyle \int_0^2 \pi\left[ \left(2x+1\right)^2-\left(x^2+1\right)^2\right]\, dx\)
7. Find the volume of the solid \(S\) whose base is the triangular region with vertices \((0,0)\), \((1,0)\) and \((0,2)\). The cross sections of \(S\) perpendicular to the \(x\)-axis are semi-circles.
\(V=\dfrac{\pi}{6}\)
8. Find the volume of the solid \(S\) whose base is the region bounded by \(y=x^2\) and \(y=2x\). The cross sections of \(S\) perpendicular to the \(y\)-axis are squares.
\(V=\dfrac{8}{15}\)
9. Find the volume of the solid \(S\) whose base is the region bounded by the parabola \(y=x^2\) and \(y=1\). The cross sections of \(S\) perpendicular to the \(y\)-axis are equilateral triangles.
\(V=\dfrac{\sqrt{3}}{2}\)