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# Week 14 Problem Set B.
Last homework assignment!
## Reading.
Chapter 5.6, (areas between curves), 6.1 (volumes using cross-sections, disk method), 6.2 (cylindrical shell method), 6.3 (arclength), 6.4 (areas of surfaces of revolution)
Also, make a one sheet cheat sheet (both sides) for the final exam.
## Problems.
### Volume of solids of revolution by cylindrical shell method.
1. In each of the following, use the shell method to find the volume of the solids generated by revolving the shaded region about the indicated axis:
1. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231201153216.png]]
2. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231201153225.png]]
2. In the following, consider the region $R$ bounded by the curves $y=x+2$ and $y=x^{2}$. Using the shell method, find the volume of the solid generated by rotating $R$ about each of the following axes:
1. The line $x=2$
2. The line $x=-1$
3. The $x$-axis
4. The line $y=4$
### Comparing washer and shell method.
Let $R$ be the region bounded by $y=x$ and $y=x^{2}$.
1. Rotate $R$ about the $x$-axis, and find the volume of this solid of revolution using the washer method.
2. Rotate $R$ about the $x$-axis, and find the volume of this solid of revolution using the shell method.
3. Rotate $R$ about the $y$-axis, and find the volume of this solid of revolution using the washer method.
4. Rotate $R$ about the $y$-axis, and find the volume of this solid of revolution using the shell method.
### Determine washer or shell method.
Consider the following shaded region $R$. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231201154249.png]]
Revolve $R$ about the $y$-axis to generate a solid of revolution. Find its volume. You may choose either shell or washer method, whichever is more convenient.
### Arclength.
Using the arclength formula to set up and compute the length of the following curves:
1. $y=(1 /3 )(x^{2} + 2)^{3 / 2}$ from $x=0$ to $x=3$.
2. $y= (x^{3} / 3) + 1 / (4 x)$ from $x = 1$ to $x = 3$.
3. $y = \int_{0}^{x} \sqrt{\sec^{4}(t)-1}\,dt$, from $x=- \pi / 4$ to $x = \pi / 4$.
4. $y=\int_{0}^{x} \sqrt{\cos 2t} \, dt$ from $x=0$ to $x= \pi / 4$.
### Surface area of surface of revolutions.
1. Find the areas of the surfaces generated by revolving the curves in each of the following about the indicated axis:
1. $y=x^{3} / 9$, where $0 \le x \le 2$, about the $x$-axis.
2. $y=\sqrt{2x-x^{2}}$, where $0.5 \le x \le 1.5$, about the $x$-axis.
2. **The surface of an astroid.** Find the area of the surface generated by revolving about the $x$-axis the portion of the astroid $x^{2 / 3} + y^{2 / 3}=1$ shown in the following diagram. Hint: Solve for $y$ as a function of $x$ first, and focus just on the part $0 \le x \le 1$, then double your result.![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231201155411.png]]
### The sphere.
In class we derived the volume and the surface area of a sphere of radius $R$. Compute them again here.
1. Derive the formula for the volume of a sphere of radius $R$ by considering the sphere generated by revolving the region bounded by $y=\sqrt{R^{2}-x^{2}}$ and $x=0$ about the $x$-axis. Use either the washer or shell method.
2. Derive the formula of the surface area of a sphere of radius $R$ by considering the surface generated by revolving the curve $y=\sqrt{R^{2}-x^{2}}$ where $-R\le x\le R$ about the $x$-axis.
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