Question

How do you find the area of the region bounded by the given curves?

`y=1/x, y=x^2, y=0, x=e`
You are suppose to use integration, but I do not know how to start the problem.

Answer 1

Graph the 4 boundaries.
Find the x coordinates of the intersect points.
Write the definite integrals for the region

Explanation:

Blue `y = x^2`

Red: `y = 1/x`

Green: `y = 0`

Orange: `x = e`

Find x coordinates of the intersect point(s):

`0 = x^2`

`x = 0`

`x^2 = 1/x`

`x^3 = 1`

`x = 1`

`x = e`

The integrals are:

`"Area" = int_0^1x^2 dx+int_1^e1/x dx`

`"Area" = [x^3/3]_0^1+ [ln(x)]_1^e`

`"Area" = 4/3`