https://file.aiquickdraw.com/mathbot/file/17267548360573upcjile.jpg

The image shows two exercises involving triple integrals over the volume of a pyramid:

### Exercice 3.7:
\[
I = \iiint_V (6xz + 2x) \, dx \, dy \, dz
\]
This integral is over the volume \( V \) of a pyramid with the following vertices:
- \( O(0, 0, 0) \),
- \( U(-2, 0, 0) \),
- \( L(0, 3, 0) \),
- \( E(2, 1, 0) \),
- \( R(0, 0, 2) \) (in 3D space).

### Exercice 3.8:
\[
J = \iiint_V \frac{dx \, dy \, dz}{(x + y + z + 1)^3}
\]
This is also an integral over the volume \( V \) of a pyramid, but with the following vertices:
- \( O(0, 0, 0) \),
- \( A(1, 0, 0) \),
- \( B(0, 1, 0) \),
- \( C(0, 0, 1) \).

Both problems require calculating triple integrals to find the volume of these pyramids.

Would you like a detailed step-by-step solution for either or both of these integrals?

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Here are 5 related questions:
1. How do you find the bounds of integration for a triple integral over a pyramid?
2. What are the common methods for calculating volumes using triple integrals?
3. How does changing the order of integration affect solving a triple integral?
4. What role do symmetries play in simplifying the evaluation of triple integrals?
5. How do you interpret a physical problem involving a triple integral over a non-rectangular region?

**Tip:** When dealing with integrals over geometric shapes like pyramids, it's helpful to sketch the region to visualize the limits of integration clearly.

Calculate the triple integral over the volume of a pyramid in exercises 3.7 and 3.8 as shown in the image.

Solve triple integrals over the volume of pyramids as seen in exercises 3.7 and 3.8. These problems involve calculating volume using multivariable calculus and integration techniques. Ideal for students in advanced calculus.

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