I was tasked to evaluate an iterated integral to find some volume and it seems that I can’t get it right.
The problem is: integral of square root of $x$ with $x$ limits $x=y^2$ to $x=2-y$ and $y$ limits $y=-2$ to $y=1$. Integrate with respect to $x$ first and then with respect to $y$.
$$\int_{-2}^1 \int_{y^2}^{2-y} \sqrt{x} dx dy$$
My answer is $\frac{323}{30}$ but the correct one should be $\frac{163}{30}$. I have used WolframAlpha integral calculator and it gave the right $\frac{163}{30}$ result. But Symbolab integral calculator gave $\frac{323}{30}$.
What is going on?
$\sqrt{x}$ has an antiderivative $\frac23 x^{3/2}$, after integration over $x$, you get $$ \int_{y^2}^{2-y} \sqrt{x} dx = \frac23\left((2-y)^{3/2} - (y^2)^{3/2}\right) = \frac32\left((2-y)^{3/2} - |y|^3\right)$$
If your symbolic calculator simplifies $(y^2)^{3/2}$ to $y^3$ without checking whether $y$ is negative, you get $\frac{323}{30}$ instead of the correct answer $\frac{163}{30}$.