Question

What is the area under the curve `f(x)=x^2+2` from 1 to 3?

Answer 1

`"area" = 38/3`

Explanation:

To determine the area under the curve for this function you need to integrate it on the interval `[-1,3]`

`"area" = F(x) = int_1^3f(x)dx`

In your case, you would get

`F(x) = int_1^3(x^2 + 2)dx = int_1^3x^2dx + 2int_1^3dx`

This is equivalent to

`F(x) = (1/3x^3 + 2x + C)|_1^3`

Evaluate this for the extremes of integration to get

`F(x) = (1/3 * 3^3 + 2 * 3 + C) - (1/3 * 1^3 - 2 * 1 + C)`

`F(x) = 9 + 6 + color(red)(cancel(color(black)(C))) - 1/3 - 2 - color(red)(cancel(color(black)(C)))`

`F(x) = (27 + 18 -1 - 6)/3 = color(green)(38/3)`