# Thread: evaluating a definite integral

1. ## evaluating a definite integral

How would I evaluate this integral using the 2nd Fundamental Theorem of Calculus? Thanks in advance.

2. Ahh I love these questions.

Let $\displaystyle \frac{2i}{n}=x$. Thus $\displaystyle dx=\frac{2}{n}$.

Now rewrite to an integral integral with this substitution.

$\displaystyle \int_{L} 1 +x +x^2dx$

Now for the bounds.

For the first bound let $\displaystyle i=0$ and for the second let $\displaystyle i=n$. So the bounds are from 0 to 2.

Got it?

3. Originally Posted by Jameson
Ahh I love these questions.

Let $\displaystyle \frac{2i}{n}=x$. Thus $\displaystyle dx=\frac{2}{n}$.

Now rewrite to an integral integral with this substitution.

$\displaystyle \int_{L} 1 +x +x^2dx$

Now for the bounds.

For the first bound let $\displaystyle i=0$ and for the second let $\displaystyle i=n$. So the bounds are from 0 to 2.

Got it?
Shouldn't that second bound be $\displaystyle \infty$?

-Dan

4. I don't believe so. Why would it be inifinity? On these problems i=0, then i=n. Is there another way you do it?

5. For the integral, the limits of x are indeed 0 to 2.

6. Yea that explanation makes sense Jameson, but I'm still not sure how this would be solved once reaching that point. How'd you determine the bounds were 0 and 2?

7. Originally Posted by thedoge
How would I evaluate this integral using the 2nd Fundamental Theorem of Calculus? Thanks in advance.

The standard form the the Riemann Ingegral is,
$\displaystyle \lim_{n\to \infty} \sum_{k=1}^n f(a+k\Delta x)\Delta x$
In this case,
$\displaystyle \Delta x=\frac{2}{n}$
Thus,
$\displaystyle b-a=2$
But there is not "a" term,
$\displaystyle a=0$ thus, $\displaystyle b=2$.
And the function is,
$\displaystyle f(x)=1+x+x^2$

Thus we need to find,
$\displaystyle \int_0^2 1+x+x^2 dx$
Which we can easily evaluate by Fundamental theorem because this function is continous on the closed interval.

8. Originally Posted by topsquark
Shouldn't that second bound be $\displaystyle \infty$?

-Dan
Momentary brain fart!

-Dan