Evaluate the triple integral of 2x - 2yz over the region E where 2 <= x <= 3, 0 <= y <= 2x - 2, and 0 <= z <= -2x + 4y - 4. Enter an exact answer.

Step-by-step solution

1. 1

   Integrate in z with x and y held fixed. Writing $Z=-2x+4y-4$ for the upper limit, $\int_0^{Z}(2x-2yz)\,dz=2xZ-yZ^2$. Only $z$ moves here, so $2x$ integrates to $2xZ$ and $-2yz$ integrates to $-yZ^2$.

2. 2

   Substitute the limit and expand. With $Z=4y-2x-4$ the inner result expands to $-16y^3+16xy^2+32y^2-4x^2y-8xy-16y-4x^2-8x$. Expanding fully now is safer than trying to keep $Z$ folded through the next integration.

3. 3

   Integrate in y from 0 to 2x - 2. Antidifferentiating term by term and substituting $y=2x-2$ gives the outer integrand $-\frac{88}{3}x^4+\frac{616}{3}x^3-528x^2+\frac{1600}{3}x-\frac{544}{3}$, which factors as $-\frac{8}{3}(x-1)\left(11x^3-66x^2+132x-68\right)$.

4. 4

   Antidifferentiate in x. $\int\left(-\frac{88}{3}x^4+\frac{616}{3}x^3-528x^2+\frac{1600}{3}x-\frac{544}{3}\right)dx=-\frac{88}{15}x^5+\frac{154}{3}x^4-176x^3+\frac{800}{3}x^2-\frac{544}{3}x$.

5. 5

   Evaluate from x = 2 to x = 3. The antiderivative gives $-\frac{466}{5}$, that is $-93.2$. Note the answer is negative because the $-2yz$ term dominates over most of the wedge.

6. 6

   Numerical check. Nested adaptive quadrature on the same limits returns $-93.200000$, agreeing with $-\frac{466}{5}=-93.2$ to six decimal places.

7. 7

   Watch the arithmetic in the y-step. A widely circulated solution to this problem reports $-\frac{4868}{45}\approx -108.2$; that value fails the numerical check, and the error enters when the expanded $y$-antiderivative is evaluated at $y=2x-2$.