Evaluate the triple integral of -5x + 5yz over the region E where 1 <= x <= 2, 0 <= y <= 5 - 5x, and 0 <= z <= -x - 2y - 5.

Step-by-step solution

1. 1

   Notice the limits are degenerate, and decide what that means. For $x$ in $(1,2]$ the stated upper $y$-limit $5-5x$ is negative, so it lies below the lower limit $0$; the same happens for $z$, whose upper limit $-x-2y-5$ is negative. The region as written is empty as a solid, so the problem is to be read as the iterated integral with those limits, where a reversed range contributes with the opposite sign.

2. 2

   Integrate in z. With $Z=-x-2y-5$, $\int_0^{Z}(-5x+5yz)\,dz=-5xZ+\frac{5y}{2}Z^2$. Substituting and expanding gives $10y^3+10xy^2+50y^2+\frac52x^2y+35xy+\frac{125}{2}y+5x^2+25x$.

3. 3

   Integrate in y from 0 to 5 - 5x. Antidifferentiating and substituting the (negative) upper limit gives the outer integrand $\frac{14125}{12}x^4-\frac{20200}{3}x^3+\frac{28425}{2}x^2-\frac{39250}{3}x+\frac{53125}{12}$.

4. 4

   Antidifferentiate in x. $\int$ of that quartic is $\frac{2825}{12}x^5-\frac{5050}{3}x^4+\frac{9475}{2}x^3-\frac{19625}{3}x^2+\frac{53125}{12}x$.

5. 5

   Evaluate from x = 1 to x = 2. The difference of the antiderivative at the endpoints is $\frac{25}{2}=12.5$. The result is positive even though the integrand $-5x+5yz$ is negative for much of the range, because the two reversed orientations flip the sign twice.

6. 6

   Numerical check. Evaluating the same iterated integral numerically (with the reversed inner ranges kept as written) returns $12.500000$, matching $\frac{25}{2}$.

7. 7

   Contrast with a common wrong value. A frequently seen worked solution reports $-\frac{40625}{72}\approx -564.2$; that value does not survive the numerical check, so the sign handling of the reversed limits must be done carefully.