Question

How do you find the area bounded by `y=x`, `y=1/x^2` the x axis, and x=3?

Answer 1

`S = 7/6`

Explanation:

Start by solving the inequality:

`x <= 1/x^2 => x^3 <= 1 => x < 1`

so we can see that in the interval `[0,1]` the area is bounded by the curve `y = x` while in the interval `[1,3]` the area is bounded by the curve `y=1/x^2`

The required area can be calculated as the definite integral of the bounding curve and therefore is:

`S = int_0^1 xdx + int_1^3 dx/x^2`

`S = [x^2/2]_0^1 - [1/x]_1^3`

`S = 1/2 - 0 -1/3 + 1 = 7/6`