Homework # 17 - Math 250
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. Integrate \( \int\int_R x dA\) where \(R = \{(x,y): x^2 + y^2 \leq 1\). (If you think about this, you can avoid a lot of work.)
a. \(-\pi\)
b. \(-\pi / 2\)
c. 0
d. \(\pi / 2\)
e. \(\pi\)
2. Integrate \( \int\int_R 1 dA\) where \(R = \{(x,y): x^2 + y^2 \leq 1, x\geq 0 \). (Again, if you think about this, you can avoid a lot of work.)
3. Integrate \( \int\int_R x + 2 dA\) where \(R = \{(x,y): x^2 + y^2 \leq 1\). (Again, if you think about this, you can avoid a lot of work.)
a. \(-2\pi\)
e. \(2\pi\)
4. Integrate \( \int\int_R e^{x^2+y^2} dA\) where \(R = \{(x,y): x^2 + y^2 \leq 1\}\).
a. \(\frac{\pi}{2}(e - 1)\)
b. \(\pi(e - 1)\)
c. \(\frac{\pi e}{2}\)
d. \(\pi e\)
e. \(\frac{\pi e}{3}\)
5. Find the volume of the solid inside the sphere \(x^2+y^2+z^2 \leq 4\), inside the cylider \( x^2 + y^2 \leq 1\), and above the xy-plane. (Hint: use the cylinder to get the 2-D region over which to integrate.)
a. \(\frac{2\pi}{3}(8-3\sqrt{3})\)
b. \(\frac{\pi}{2}(8-3\sqrt{3})\)
c. \(\frac{2\pi}{5}\)
d. \(\frac{2\pi}{5}(8-3\sqrt{3})\)
e. \(\frac{\pi}{5}(8-3\sqrt{3})\)
In the next three problems, use the laminum on the region \(R = \{(x,y): x+y \leq 1, x\geq 0, y \geq 0\}\), where distance is measured in meters, with density function \(\rho(x,y) = x + y + 1\) kg/m. (I highly recommend that you sketch this region for yourself.)
6. Find the total mass of this laminum.
a. \(\frac{1}{6}\) kg
b. \(\frac{1}{2}\) kg
c. \(\frac{2}{3}\) kg
d. \(\frac{5}{6}\) kg
e. \(1\) kg
7. Find the x-coordinate of the center of mass \(\overline{x}\).
a. \(\frac{1}{4}\) m
b. \(\frac{3}{10}\) m
c. \(\frac{7}{20}\) m
d. \(\frac{2}{5}\) m
e. \(\frac{9}{20}\) m
8. Find the y-coordinate of the center of mass \(\overline{y}\). (If you think about the region and the density function, you can save yourself some work with this one.)