9Triple integrals PPT

Triple Integrals Instructor by Assist.Prof.Siriwan Wasukree

Final Exam MAT137 Time: 09:30 - 12:00 AM , Thursday 29 April 2021. The exam includes the following topics 1—9 (40 points) 1. Derivatives of Vector Function (4 points) 2. Find gradient divergence and curl of vector functions (4 points) 3. Integrals of Vector Function (4 points) 4. Find the work (4 points) 5. Double integrals (4 points) 6. Use Double integrals to find the area (5 points) 7. Use polar coordinates to find Double integrals (5 points) 8. Triple integrals (5 points) 9. Use Triple Integrals to find the volume (5 points)

Regulation: 1. Online exams using Google meet. 2. You must open the video while taking the exam. 3. You can look at the documents during the exam, but be careful of not taking the exam in time. 4. When the exam time, I sent a link to each question on google form into each student’s LINE. 5. You must prepare a piece of paper to write all your answers and you must write your student ID and surname at the top right corner of every sheet. 6. When you finish each question, then take a picture of your answer sheet clearly to upload in google form. 7. The online system is open to start doing and when the time is up, the system immediately shuts down. 8. You should submit an answer before the time expires. If you wait until the system runs out and the system shuts down, you may receive 0 points. 9. You must be honest with yourself and others.

Triple Integrals List of Topics in this class 1. Definition of Triple Integrals 2. Volume by Triple Integrals

1. Definition of Triple Integrals A Double Integrals f (x, y)dA is evaluated R over a two dimensional region in the plane. The same way, a Triple Integrals f (x, y, z)dV is evaluated S over a three dimensional region in the space.

1. Definition of Triple Integrals Fubini’s Theorem over a parallelepiped in space If ƒ(x, y, z) is continuous over the rectangular solid S: a  x  b , c  y  d , r  z  s then the Triple Integrals may be evaluated by the iterated integral  f (x, y, z)dV  s d b f (x, y, z)dxdydz S r c a The iterated integration can be performed in any order (with appropriate adjustments to the limits of integration): dxdydz dxdzdy dzdxdy dydxdz dydzdx dzdydx

Example 1 Evaluate z2yexdV where S is the box given by S 0  x  1 , 1  y  2 , 1  z  1

Example 1 (Cont.)

  Example 2 Evaluate23 6xy dzdydx 2 0 0

2. Volume by Triple Integrals Volume of a Region in Space The volume of the three-dimensional region S in space is V  dV S

Example 3 Find the volume of the region that lies behind the plane x + y + z = 8 and in front the region in the yz-plane that is bounded by z  3 y and z  3 y 2 4

Example 3 (Cont.)

Example 4 Use Triple Integrals to find the volume of the region between z = x +y , y = 2x , z = 0 , y = 2 and x = 0 Limit of z is Limit of y is Limit of x is

Example 4 (Cont.)

Activity Today  Do activity now.  Send to album in line group.  When you’re done, you can exit the meet room.

Activity today Use Triple Integrals to find the volume of the region between the cylinder z = y2 and the xy-plane that is bounded by the planes x = 0, x = 1, y = 1, y = 1  Do activity now.  Send to album in line group.  When you’re done, you can exit the meet room.

See you next week