Question

What is the net area between `f(x) = 3-xsqrt(x^2-1)` and the x-axis over `x in [2, 3 ]`?

Answer 1

The area `~~2.81`

Explanation:

Here is a graph of the function `f(x) = 3-xsqrt(x^2-1)`:

graph{3-xsqrt(x^2-1) [-1, 5, -7, 3]}

Please observe that the `f(x)` is negative over the region `[2,3]`, therefore, to obtain an positive area, we shall evaluate from 3 to 2.

To integrate `-xsqrt(x^2-1)dx` let `u =x^2-1` and `du = 2xdx`

`intx(x^2-1)^(1/2)dx = int1/2u^(1/2)du = 2/3(1/2)u^(3/2) = 1/3u^(3/2)`:

The integration of the constant term is trivial:

`int_3^2 3-xsqrt(x^2-1)dx = (3x - 1/3(x^2-1)^(3/2)]_3^2=`

`3(2)-1/3(2^2-1)^(3/2)-3(3)+1/3(3^2-1)^(3/2) ~~2.81`