How do I integrate e^(y^3) in a definite integral without using erf(x)?

[exidez]

The function i need to integrate is

$$\displaystyle\int^3_0 \int^1_{\sqrt{\frac{x}{3}}} e^y^3,dydx$$

however my problem is integrating:
$$\int^1_{\sqrt{\frac{x}{3}}} e^y^3 dy$$

I have looked on the forum and everyone is mentioning erf(x) which i don't understand yet. Do i need this for this definite integral? If so where can i learn about it and how to use it?

 

[HallsofIvy]

Try changing the order of integration.

For this integral, x ranges from 0 to 3 and, for each x, y from $\sqrt{x/3}$.

When x= 3, $y= \sqrt{x/3}= 1$ and when x= 0, $y= \sqrt{0/3}= 0$ so y ranges from 0 to 1. For each y, since $y= \sqrt{x/3}$ leads to $y^2= x/3$ or $x= 3y^2$, x ranges from 0 to 3y^2. Try
$$\int_{y= 0}^1 \int_{x= 0}^{3y^2} e^{y^3} dx dy$$

 

[exidez]

that actually worked out quite nicely. Thank you.