Week 5

Tutorial 5: Volume Integrals

Question PDF

Tutorial 5 Volume integrals.pdf

374.8KB

Class Recording

kya-habw-nht (2022-04-14 20_00 GMT-7).mp4

Whiteboard

Week5 - Google Jamboard

The version of the browser you are using is no longer supported. Please upgrade to a supported browser. Dismiss

https://jamboard.google.com/d/1Om_7eOKAmq_Tgy9jL4MQBeOgykFwLRG_8MZZDss6snM/edit?usp=sharing

1. Find a volume bounded by the surface , and, .
👉
Solution

2. Find the total mass of a spherical object with radius of and with a density function of , where is the radial distance from the center of the sphere.
👉
Solution

3. Find the moment of inertia of the following hollow cylinder with constant density along the -axis.



👉
Solution

4. Find the volume of the following paraboloid. Hint: Inside of the cone boundary is not empty, but it is used to calculate the height upper boundary of the paraboloid ().



👉
Solution
Radius upper boundary:
Height upper boundary:
Hence, the cylindrical coordinate boundary is: , ,

5. Evaluate where is enclosed by , , and .
👉
Solution
We will use cylindrical coordinates to easily solve this sum. Converting the given outer limits of we get:
Since there is no limitation on the values of , we assume it has the values of to . This is also in accordance with the diagram which is obtained by drawing these surfaces.
Also, we substitute in the argument of the integral.
Thus, the integral is,

6. Integrate the function over the volume enclose by the planes and , and between the surfaces and .
👉
Solution
Since is expressed as a function of , we should integrate in the direction first. After this we consider the -plane.
The two curves meet at and . We can integrate in or  first. If we choose , we can see that the region begins at and ends at , and so is between 0 and 1.
Thus, the integral is,

7. Find the volume of the tetrahedron with corners at , , , and .
👉
Solution
The whole problem comes down to correctly describing the region by inequalities:
The lower limit comes from the equation of the line that forms one edge of the tetrahedron in the x-y plane
The upper limit comes from the equation of the plane that forms the “upper” side of the tetrahedron. Now the volume is

8. Find the volume under above the quarter circle inside in the first quadrant.
👉
Solution

9. An object occupies the space inside both the cylinder and the sphere , and has density at . Find the total mass.
👉
Solution

10. Find the volume bounded by the cylinder and the planes ; .



👉
Solution
Place the volume element in general position, and integrate to a rod along the -direction. Then integrate the rod over the circle: