# Find $\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$

$$I:=\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$$ Here the region of integration is the triangle with vertices $$(0,0),(0,1)$$ and $$(1,1)$$ and given as a type-1 region. We can convert it into a type-2 region which makes the integral easier. $$I=\int_{0}^{1}\int_{0}^{y}y^4e^{xy^2}dx dy=\int_{0}^{1} y^2(e^{y^3}-1)dy=\frac {e-2}{3}$$ Is this correct? I'd like to add graphs but I'm still learning how to do that.

• Yes, you have it. Well done. – Mark Viola Jan 30 at 4:37