Not sure if this surface integral is right

In summary, the conversation discusses a manual integration and surface calculations for a prism. The individual encountered some errors in their calculations, but with the help of another person, they were able to correct their mistakes and get the final answer. The main errors were related to the signs of the normal and the direction of the faces.
  • #1
unscientific
1,734
13

Homework Statement


16jqcw.png


The Attempt at a Solution



I did the manual integration of part (i) and got an answer of 5/6 instead, I'm not sure which part is wrong.. For the surfaces, I start off with the surface in the x-z plane, then the slanted plane, then the y-z plane, then the top of the prism and finally the bottom of the prism.

For the slanted plane, I projected it onto the x-z plane to do the integration..

Part (a)
2dtngx0.png

k2beox.png
 
Last edited:
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  • #2
I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy2+x2, but went wrong in the next line.
 
  • #3
haruspex said:
I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy2+x2, but went wrong in the next line.

Thanks! I will try again.
 
  • #4
haruspex said:
I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy2+x2, but went wrong in the next line.

Finally got the answer as 1/4 ! Thanks alot!
 

FAQ: Not sure if this surface integral is right

1. How do I determine if my surface integral is correct?

To determine the correctness of a surface integral, you should first check if you have correctly set up the integral using the appropriate surface element and limits of integration. Then, you can evaluate the integral and compare it to the known solution or use a calculator to check for any errors in your calculations.

2. What are the common mistakes in solving surface integrals?

Some common mistakes in solving surface integrals include using the wrong surface element, using incorrect limits of integration, and making calculation errors. It is also important to pay attention to the orientation of the surface and the direction of the normal vector when setting up the integral.

3. How can I verify the accuracy of my surface integral?

To verify the accuracy of your surface integral, you can compare it to the known solution or use a calculator to check for any errors. You can also try using a different method, such as parametrization or using a different coordinate system, to solve the integral and see if you get the same result.

4. What are the applications of surface integrals in science?

Surface integrals have various applications in science, including in physics, engineering, and mathematics. They are used to calculate the flux of a vector field through a surface, the surface area of a three-dimensional object, and the mass of a curved object. They are also used in fluid mechanics, electromagnetism, and differential geometry.

5. Are there any tips for solving surface integrals more efficiently?

Some tips for solving surface integrals more efficiently include carefully choosing the surface element and limits of integration, paying attention to the orientation of the surface and the direction of the normal vector, and breaking the surface into smaller, simpler pieces if possible. It can also be helpful to practice and familiarize yourself with different techniques, such as parametrization and using different coordinate systems, for solving surface integrals.

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