# How should I calculate $\int_0^1\int_{x^2}^{x}xy(x+y)\, dy dx$?

The question asks me to calculate the integral of the function $xy(x+y)$ over the region bounded by the curves $y=x$ and $y=x^2$. I understand that this is a double integral, and when I calculate the double integral, it appears as $$\int_0^1\int_{x^2}^{x}xy(x+y) dy dx$$ I get the answer $3/56$, but the answer in the text as $\pi/4$. Could someone point out what am I doing wrong here. Thank you very much.

• Show your work so that someone can point out your mistake. – Sahiba Arora Jun 2 '17 at 19:47
• I don't see how $\pi$ could be included in the answer. I believe your integral representation is correct. – Kaynex Jun 2 '17 at 19:52
• The integral in the question do equal to $\frac{3}{56}$. It is very likely your copy down the wrong integral. You should include the "exact statement of the question" you are dealing with in this question. – achille hui Jun 2 '17 at 19:54
• That's why I ask you to include the exact statement of the question so that we can figure out whether you are misinterpreting some part of the question or the text you refer to is simply wrong. BTW, the double integral you write down is correct. – achille hui Jun 2 '17 at 20:03
• Textbooks do occasionally make mistakes. It's possible the author omitted a square root symbol, for example, or interchanged two problems (so you could look to see if there is a nearby problem for which the text gives $3/56$ as the answer). Or $\pi/4$ could be the correct answer to a completely different problem in an earlier edition, and only the problem got changed.. – Barry Cipra Jun 2 '17 at 20:05

Your solution is correct. Before we even to that, just look at it. Where the hell would a $\pi$ come from? Now, beyond that, I carried out the integral in probably the same way that you did and got the same answer. Moreover, I verified it here: WolframAlpha.