Question

Evaluate the triple integral ∫∫∫_b xyz² dv, where b is the rectangular box.

B= {(x,y,z)∣0≤x≤1,−1≤y≤7,0≤z≤4} integrating first with respect to y, then z, and then x.

Answer 1

The volume under the graph of the function xyz² within the specified rectangular box is obtained by performing a triple integral, integrating first regarding y, then z, and finally x.

Step-by-step explanation:

To evaluate the triple integral ∫∫∫_b xyz² dv where b is the rectangular box B= 0≤x≤1, −1≤y≤7, 0≤z≤4, we integrate first with respect to y, then z, and then x. Since the limits of integration are constants, we can separate the integral into the product of three definite integrals:

∫∫∫_b xyz² dv = ∫x dx ∫∫ yz² dy dz

The integration proceeds as follows:

1. Integrate with respect to y from −1 to 7, which will result in a function of x and z that includes a polynomial in z.
2. Then integrate this result with respect to z from 0 to 4, obtaining a polynomial in x.
3. Finally, integrate the resulting function with respect to x from 0 to 1.

This process will yield the numerical value of the integral, representing the volume under the graph of the function xyz² within the rectangular box B.