Solved: Definite Integrals - Answers & Explanations

[lLovePhysics]

[SOLVED] Definite Integrals

Homework Statement

$$\int_{1}^{3}x^{2}dx$$

Homework Equations

The Attempt at a Solution

Why is the answer 26/3? I got 4 by using the limit/Riemann Sum definition. Is this one method to calculate definite integrals?

 

[rocomath]

by just taking the anti-derivative, i get 26/3. check your work again.

 

[lLovePhysics]

Sorry, but I don't think I can use any other methods because of my teacher.

I got 2n for the width of the rectangle and 1+(2i/n) for the height..

AHHHH: I found my error.. I forgot to square the 2i/n -_____-

 

[lLovePhysics]

Oh jesus christ... My answer never matches. Perhaps I should do everything step-by-step?

 

[lLovePhysics]

by just taking the anti-derivative, i get 26/3. check your work again.

Thanks a lot for bothering to post those other methods. I hope I will get to learn/use them later!

 

[aq1q]

hey, show some work. But if u are stuck as where to start.. do these
find Delta X, find Xi. and then use this formula

lim $$\sum f(Xi)\Delta X$$
n $$\rightarrow \infty$$

ugh that's so bad... i have never had to use this.. but like..i'm sure u can find it on wikipedia
In addition, know what the summation of a X^2 series is.. it is n(n+1)(2n+1)6\

ah, i feel so bad.. i couldn't draw it successfully.

 

[aq1q]

so basically it is
2/n$$\sum (1+2i/n)^2$$

then use foil(or whatever u call it).

it becomes (1+4i/n +4i^2/n)

 

[lLovePhysics]

Yeah, thanks aq1q. That's what I did but I didn't square the x term for the height. I also messed up on simplifying. I just make too many mistakes :/

 

[aq1q]

ah :\ so everything ok now?

 

[Dietrick]

Alternatively:

Take the integral:

[(1/3)(x)^3] over 3 and 1

Then evaluate:

[(1/3)(3)^3] - [(1/3)(1)^3] = 26/3