Question

How do you evaluate the definite integral `int (x^3-pix^2 dx` from `[2,5]`?

Answer 1

a

Explanation:

These steps should all be pretty understandable for you, but if not, check this out

First, find the indefinite integral
`int(x^3-pix^2)dx=int(x^3)dx-int(pix^2)dx`
`=int(x^3)dx-pi int(x^2)dx`
`=(x^4/4)-pi(x^3/3)`

If you didn't understand that last step, this may help.

Now that we have the indefinite integral, we can use (one part of) the fundamental theorem of calculus to solve the definite integral.

`int_2^5(x^3-pix^2)dx= [(x^4/4)-pi(x^3/3)]_2^5`
`=[(5^4/4)-pi(5^3/3)]-[(2^4/4)-pi(2^3/3)]`
`=[(625/4)-pi(125/3)]-[(16/4)-pi(8/3)]`
`=(625/4)-pi(125/3)-(16/4)+pi(8/3)`
`=((625-16)/4)-pi((125+8)/3)`
`=609/4-pi133/3`
This is exact, but we can approximate if you want
`=12.9727257`