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# Week 14 Problem Set A.
## Reading.
Chapter 5.6, (areas between curves), 6.1 (volumes using cross-sections, disk method), 6.2 (cylindrical shell method), 6.3 (arclength)
## Problems
### Areas.
In each of the following, find the geometric area of the shaded regions.
1. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121310.png]]
2. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121322.png]]
3. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121333.png]]
4. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121341.png]]
5. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121441.png]]
6. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121455.png]]
7. ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121509.png]]
8. Sometimes, you want to integrate with respect to the $y$ variable instead. Find the area of this shaded region:![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121606.png]]
9. And remember to chop up the regions if necessary. Find the area of the shaded region:![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129121717.png]]
If a diagram is not provided to you, then you have to sketch it out first. Be careful of noting where are the intersections.
1. Find the areas of the regions enclosed by the lines and curves in each of the following.
1. $y = x^{2}-2$ and $y=2$
2. $y=x^{4}$ and $y=8x$
3. $y=x^{2}$ and $y=-x^{2}+4x$
4. $y=\sqrt{|x|}$ and $5y=x+6$. Hint: How many intersections are there?
5. $4x^{2}+y=4$ and $x^{4}-y=1$
6. $y=2\sin(x)$ and $y=\sin(2x)$ over the interval $0\le x \le \pi$.
7. $y=\sec^{2}(x)$ and $y=\tan^{2}(x)$, $\displaystyle x=-\frac{\pi}{4}$ and $\displaystyle x=\frac{\pi}{4}$
### Volume of revolution by disk method.
In each of the following, find the volume of the solid generated by revolving the shaded region about the given axis.
1. About the $x$-axis: ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129122431.png]]
2. About the $y$-axis: ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129122451.png]]
3. About the $x$-axis: ![[1 teaching/smc-fall-2023-math-7/week-14/---files/Pasted image 20231129122505.png]]
Find the volumes of the solids generated by revolving the region bounded by the lines and curves in each of the following about the $x$-axis:
1. $y=x^{2}$, $y=0$, $x=2$
2. $y=\sqrt{9-x^{2}}$, $y=0$
3. $y=\sqrt{\cos x}$, $0\le x \le \pi / 2$
4. $y = \sec x$, $y=0$, $x = -\pi / 4$, $x = \pi / 4$.
We can also revolve the region about different axes to generated different solids of revolution. Modify the method to solve the following:
Find the volume of the solid generated by revolving the region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about each of the following axes:
1. the $x$-axis
2. the $y$-axis
3. the line $y=2$
4. the line $x=4$.
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