Question

To evaluate the integral of .

Short answer

The value of the given iterated integral is \(\frac{{27}}{2}\).

Step-by-step solution

The given function and Concept of Fubini’s theorem for triple integrals

The function is \(f(x,y,z) = y\).

The region is \(E = \{ (x,y,z)\mid 0 \le x \le 3,0 \le y \le x,x - y \le z \le x + y\} \).

Fubini’s theorem for triple integrals:

If\(f\)is continuous on the rectangular box\(B = (a,b) \times (c,d) \times (r,s)\), then

Integrate the given integral with respect to \(z\) and apply the limit of it

The given integral is .

Integrate the given integral with respect to \(y\) and apply the limit

On Integrating the given interval with respect to \(y\) and applying the limit:

Integrate the given integral with respect to \(x\) and apply the limit

On integrating the given interval with \(x\) and applying the limit:

Thus, the value of the given iterated integral is \(\frac{{27}}{2}\).