Question

Find the area between the graph of the given function and the x-axis over the given interval, if possible.

f(x)=3/(x-1)^3, for [-[infinity], 0]

Answer 1

`A= -(3)/(2)`

area between curve and the x-axis, within the intervals
`x=[-\infty,0)`

Explanation:

Given function is:

`f(x) = (x)/((x-1)^3)`

to find its area between the intervals
`x=[-\infty,0)`, we'll need to integrate it.

`A = \displaystyle{\int^0_(-\infty) {(x)/((x-1)^3) \, dx}`

`A = \left|- (3)/(2 \left(x - 1\right)^(2))\right|^0_(-\infty)`

`A = \left(- (3)/(2 \left(0 - 1\right)^(2))\right)-\left(- (3)/(2 \left(-\infty - 1\right)^(2))\right)`

`A= \left(- (3)/(2)\right)-0`

`A= -(3)/(2)\,\text{unit}^2`