Final answer:
To evaluate the triple integral of (xy + xz + yz) over the given region, we integrate with respect to one variable at a time, taking advantage of the polynomial nature of the integrand and the symmetry of the bounds. After three steps, the value of the integral is found to be 560.
Step-by-step explanation:
The student is asking to evaluate a triple integral of the function (xy + xz + yz) over a rectangular box with the ranges -5 ≤ x ≤ 5, -2 ≤ y ≤ 2, and -7 ≤ z ≤ 7. To solve this, we can integrate each term separately because the integral of a sum is the sum of the integrals. The triple integral can be broken down into three separate double integrals due to the independent limits of integration for x, y, and z.
We will first integrate with respect to x, then y, and finally z. We perform the integral by treating each variable that is not currently the variable of integration as a constant. Since there are no functions more complex than polynomials and given the symmetry of the problem (with respect to the x and y variables), we can simplify the process.
Step 1: Integrate with respect to x:
- ∫ (∫_{-5}^{5} x dx) y + ∫ (∫_{-5}^{5} x dx) z + ∫ yz dx = 0 * y + 0 * z + 10yz
Step 2: Integrate with respect to y:
- ∫ 10yz dy from -2 to 2 = 40z
Step 3: Integrate with respect to z:
- ∫ 40z dz from -7 to 7 = 560
Therefore, the value of the integral is 560.