Math Problem Statement

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

Solution

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

Exercice 3.7:

I=V(6xz+2x)dxdydzI = \iiint_V (6xz + 2x) \, dx \, dy \, dz This integral is over the volume VV of a pyramid with the following vertices:

  • O(0,0,0)O(0, 0, 0),
  • U(2,0,0)U(-2, 0, 0),
  • L(0,3,0)L(0, 3, 0),
  • E(2,1,0)E(2, 1, 0),
  • R(0,0,2)R(0, 0, 2) (in 3D space).

Exercice 3.8:

J=Vdxdydz(x+y+z+1)3J = \iiint_V \frac{dx \, dy \, dz}{(x + y + z + 1)^3} This is also an integral over the volume VV of a pyramid, but with the following vertices:

  • O(0,0,0)O(0, 0, 0),
  • A(1,0,0)A(1, 0, 0),
  • B(0,1,0)B(0, 1, 0),
  • C(0,0,1)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?


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.

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Math Problem Analysis

Mathematical Concepts

Triple Integrals
Volume of Pyramids
Multivariable Calculus

Formulas

Triple Integral: I = ∭_V (6xz + 2x) dxdydz
Triple Integral: J = ∭_V dxdydz / (x + y + z + 1)^3

Theorems

Multivariable Integration Theorem
Volume Calculation using Triple Integrals

Suitable Grade Level

Undergraduate Level (Calculus III)