Question

Evaluate ∭(B) ze^x+ydV where B is the box bounded by 0≤x≤5,
0≤y≤2, and 0≤z≤1. The value is ___.

Answer 1

To evaluate the triple integral over the box B for the function ze^x+ydV, one needs to perform three separate integrals over the respective variables z, y, and x within their specified bounds, and multiply the results together to obtain the final answer.

Step-by-step explanation:

The question asks for the evaluation of a triple integral over a rectangular box B, which represents a volume in three-dimensional space. The given integral is ∫∫∫(B) ze^x+ydV, and the limits for the integration are from 0 to 5 for x, from 0 to 2 for y, and from 0 to 1 for z. To solve this, we perform the triple integral by integrating first with respect to z, then y, and finally x, while considering that the integrand can be separated into a product of functions, each in a single variable.

First, integrate with respect to z: ∫_0^1 z dz = \frac{1}{2}z^2 |_0^1 = \frac{1}{2}.

Then, integrate with respect to y: ∫_0^2 e^y dy = e^y |_0^2 = e^2 - 1.

Now, integrate with respect to x: ∫_0^5 e^x dx = e^x |_0^5 = e^5 - 1.

The final result is obtained by multiplying the results of the individual integrals: (\frac{1}{2})(e^2-1)(e^5-1).