Question

Integration

1. Using the Laplace transform pairs in the formula sheet (available in the Start Here module on the Additional Resources page), derive the Laplace transforms for the following time functions. For full credit, be sure to specify the theorem you will use and the function for which you will use the Laplace Transform.

\text { a. } e^{-a t} \cos (\omega t) u(t)

Verified

### Question 42010

Integration

Convert the equation into spherical coordinates.
\rho=18 \sec (\varphi)
\rho=18 \sin (\varphi)
x^{2}+y^{2}+(z-9)^{2}=81
\rho=\sqrt{18}
\rho=18 \cos (\varphi)

### Question 42009

Integration

\left(\sqrt{26}, \frac{\pi}{6}, \frac{\pi}{3}\right)
Convert the spherical point (p, o, 0) into rectangular coordinates.

### Question 42008

Integration

Convert the spherical point (p, q,0) into rectangular coordinates.

### Question 42007

Integration

Set up and evaluate the indicated triple integral in the appropriate coordinate system. Enter an exactanswer. Do not use a decimal approximation.
\iiint_{Q} z d V, \text { where } Q \text { is the region between } z=\sqrt{x^{2}+y^{2}} \text { and } z=\sqrt{16-x^{2}-y^{2}}
\iiint_{Q} z d V=

### Question 42006

Integration

After set up, evaluate the indicated triple integral in the appropriate coordinate system. Enter an exactanswer. Do not use a decimal approximation.
\iiint_{Q} z e^{f(x, y)} d V, f(x, y)=\sqrt{x^{2}+y^{2}}, \text { where } Q \text { is the region inside } x^{2}+y^{2}=100, \text { outside } x^{2}+y^{2}=64
and between z=0 and z=5.
\iiint_{Q} z e^{f(x, y)} d V=

### Question 42005

Integration

\text { Set up the triple integral } \iiint_{Q} f(x, y, z) d V \text { in cylindrical coordinates. }
Q \text { is the region bounded by } y=36-x^{2}-z^{2} \text { and } y=3
\int_{0}^{6} \int_{3}^{36-r^{2}} \int_{0}^{2 \pi} f(r \cos (\theta), y, r \sin (\theta)) \cdot r d y d r d \theta
\int_{0}^{36} \int_{3}^{36-r^{2}} \int_{0}^{2 \pi} f(r \cos (\theta), y, r \sin (\theta)) \cdot r d y d r d \theta
\int_{0}^{6} \int_{0}^{2 \pi} \int_{3}^{36-r^{2}} f(r \cos (\theta), y, r \sin (\theta)) \cdot r d y d r d \theta
\int_{0}^{2 \pi} \int_{0}^{6} \int_{3}^{36-r^{2}} f(r \cos (\theta), y, r \sin (\theta)) \cdot r d y d r d \theta

### Question 42004

Integration

\text { Set up the triple integral } \iiint_{Q} f(x, y, z) d V \text { in cylindrical coordinates. }
Q \text { is the region above } z=\sqrt{x^{2}+y^{2}} \text { and below } z=\sqrt{1352-x^{2}-y^{2}} .
\int_{0}^{2 \pi} \int_{0}^{26} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) d z d r d \theta
\int_{0}^{2 \pi} \int_{0}^{26} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) \cdot r d z d r d \theta
\int_{0}^{2 \pi} \int_{0}^{676} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) d z d r d \theta
\int_{0}^{2 \pi} \int_{0}^{676} \int_{\nu}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) \cdot r d z d r d \theta

### Question 42003

Integration

Write the given equation in Cylindrical coordinates.
(x-95)^{2}+y^{2}=9,025
r=95 \sin (\theta)
r=95 \cos (\theta)
r=190 \sin (\theta)
r=190 \cos (\theta)

### Question 42002

Integration

Write the given equation in cylindrical coordinates.
x^{2}+y^{2}=196

### Question 42001

Integration

Find the mass of the solid with density p(x, y, z) and the given shape.
\rho(x, y, z)=41, \text { solid bounded by } z=x^{2}+y^{2} \text { and } z=9
Mass