# Area under y=x^2

1. Sep 15, 2008

### winston2020

This is out of my textbook:

EXAMPLE 6:
Find the area under the parabola y=x2 from 0 to 1.

SOLUTION:
An antiderivative of f(x) = x2 is F(x) = 1/3x3. The required area is found using Part 2 of the Fundamental Theorem....

My question is: how was the antiderivative obtained?

2. Sep 15, 2008

### Feldoh

The second part of the fundamental theorem of calculus says, essentially, that if we have a continuous function f over some interval, [a,b]. Then let F be an antiderivative of f.

So it follows that:
f(x) = F'(x)

-----

Take the derivative of F(x) and you will indeed obtain f(x). All that happened was the derivative in reverse.

3. Sep 15, 2008

### StingerManB

When you have a variable (let's use x) you can use this formula:

(1/n+1)x^n+1.
Read as: "One over n plus one, times the variable, that variable raised to n plus one. "n" is the original power from the problem.
Example: x^4 would be 1/5 x^5. To prove this, you could find the derivative of my solution. (which would be 5*(1/5) x^5-1 = x^4.
Hope this helped a little.

4. Sep 15, 2008

Basically, to find an antiderivative, you have to think up a function that, if you take its derivative, you would get your original function back again.

Now, the Power Rule of derivative says that if $$f(x) = ax^n$$, then $$f'(x) = anx^{n-1}$$. So use this backwards:

You have $$f(x) = x^2$$. Your task is to find $$F(x)$$.

In this case, you the exponent in $$f(x)$$ is $$n-1 = 2$$, so the exponent in $$F(x)$$ is $$n = 3$$.

You also know that in $$f(x)$$, $$an = 1$$, so $$a = 1/3$$.

Now, you can write $$F(x) = ax^n = 1/3x^3$$.

5. Sep 15, 2008

### winston2020

Thanks everyone. I actually do understand how to get this particular example's anitderivative. I wanted to know if there was a more general way of achieving this. I can only imagine once f(x) gets a little more complicated, finding the antiderivative could get quite painful (although finding the derivative can also be quite painful ).

6. Sep 15, 2008

there is four way to antiderivative (as i have known ) ...

first one is subtution
second using the table of intgration
third is by part
fourth is friction

excuse my splling