Question

When you take a definite integral using normal integration techniques, do you get an exact answer or an approximation?

Answer 1

Define "normal techniques"

Answer 2

Like a simple problem, say integrating x^2 from 0 to 2. Is 4 an approximate answer or exact?

Answer 3

if you integrate x^2 from 0 to 2 you get 8/3. Maybe that is exact... lol

Answer 4

4 would be approximate.

Answer 5

if that is the question.

Answer 6

If you integrate according to the rules, such as integral of
x^n = [1/(n+1)] x^(n+1) you get an exact expression that can be evaluated later exactly between any two finite limits.

Answer 7

Haha oh wow I'm sorry, my skills are very rusty. I think I took the derivative then plugged in 2 and 0 as if it were an integral. Umm..yea no is 8/3 exact or approximate though?

Answer 8

Ok so because it's bounds are finite, your answer is exact?

Answer 9

Yeah, I think the point of the integral being as n -> infinity, means you're finding exact area.

Answer 10

But what about the dx? Aren't we approximating the area under the curve by dividing it into infinitely small pieces, each of which are approximately equal to the area under the curve at any given point?

Answer 11

Infinitely small means all the area is accounted for, exactly.

Answer 12

Ok, like infinitely small like a point then? That makes a lot of sense, thank you.

Answer 13

^yes essentially. But, strips not points. It gets more exact as you use more strips, so when you use "infinite" strips, it is exact.

Answer 14

Yea, that's what I was trying to say. You made that much clearer for me, thanks so much.