By integrating, find the area between the curve and x axis of y = x*exp(x) between x = 0 and x = 1

The area under a curve is found by integration.
Area = integral{xexp(x) dx} with limits 0 and 1
It's necessary to use integration by parts to find this integral as there are two x functions multiplied together. The formula is integral{u dv}= uv - integral{v du}.
To make this applicable to our function we use the substitution of u = x and dv = exp(x). This is done as, in general, we chose u to be the least complicated function which in this case is x. There exists a more rigourous set of rules we can follow to chose u and dv, but for this question we just need to know that u needs to get simpler when we diffrentiate it and dv musn't get more complicated when we integrate it.
==> u = x => du = 1 dx and dv = exp(x) dx => v = exp(x)
==> Area =[ xexp(x) - integral{exp(x)1 dx} ]with limits 0 and 1
==> Area = [xexp(x) - exp(x)] with limits 0 and 1
Now we need to apply limits:==> Area = [1*exp(1) - exp(1)] - [0 - exp(0)]
Remembering that exp(0) = 1 the area under the curve is:==> Area = 1