Hi, I have been having trouble finding areas of region bounded by graphs.

Like for example:

y=xe^{-(x^2/2)}, y=0, x=0, x=√2

and

y=3^{x}, y=0, x=0, and x=3

So for example, the second one.

First I would graph it.

Then I would find out the axis it gets cut off right? From which x it starts, and the x it ends.

So then it would be [0, 3]

Then I would use the y=3^{x}right? As the integral.

Then it would be (1/(3-0))∫3^{x}

=1/3∫3^{x}

Then I would find the antiderivative?

Which would equal 1/ln3 * 3^{x}= 3^{x}/ln3

Then I would plug in the points 3 and 0 right and subtract it from each other?

So 1/3[(3^{(3)}/ln3)-3^{(0)}/ln3]

Which is kind of not a pretty number... but is this the right process?

Just making sure before I do the first one... because the antiderivative of xe^{-(x^2/2)}looks a bit difficult.

So can anyone make sure if these steps are right? Thanks!

And if it really is wrong, explain the process or some advices as well? Thanks!