Thread: Using limits to evaluate integral?

1. Using limits to evaluate integral?

Use limits to evaluate integral

y=(x+2) from x=1 to x=4

I know delta x is 4/n and xi is 4i/n

my issue is with the substitution

(4i/n)+2 times (4/n) how would I do this part

2. Re: Using limits to evaluate integral?

You can't evaluate $\displaystyle \displaystyle \int_1^4{x + 2\,dx}$?

3. Re: Using limits to evaluate integral?

I am not sure my textbook just shows in the example using limits. Right now technically I am not learning calculus only an "introduction to calculus" so I have not been taught that yet.

I have only been taught to integrate using formula such as n(n+1)/2 for 1+2+3+n

4. Re: Using limits to evaluate integral?

Originally Posted by homeylova223
Use limits to evaluate integral

y=(x+2) from x=1 to x=4

I know delta x is 4/n and xi is 4i/n

my issue is with the substitution

(4i/n)+2 times (4/n) how would I do this part
If you must use Riemann Sums, first of all, your distance is 3, not 4, so the size of each interval is actually $\displaystyle \displaystyle \frac{3}{n}$, where n is the number of intervals. This is the length of each of your rectangles.

What is the width of each of your rectangles?

5. Re: Using limits to evaluate integral?

Originally Posted by homeylova223
Use limits to evaluate integral

y=(x+2) from x=1 to x=4

I know delta x is 4/n and xi is 4i/n

my issue is with the substitution

(4i/n)+2 times (4/n) how would I do this part
Learn from every problem you solve...

http://www.mathhelpforum.com/math-he...tml#post663611

6. Re: Using limits to evaluate integral?

if we chop the interval [a,b] into n pieces, each piece has length (b-a)/n. so the k-th out of n subintervals is the interval:

[a + (b-a)(k-1)/n, a + (b-a)k/n]. in this example a = 1, and b = 4. so we have the intervals:

[1,1+3/n], [1+3/n, 1+ 6/n],....,[1+(3k-3)/n, 1+3k/n],.....,[1+(3n-3)/n, 4].

now, to pick a value for xk, we need to decide how we're going to do it. let's go with the right-hand endpoint of each interval

(since f(x) = x+2 is increasing, this will be a slightly high estimate for the integral).

this gives us:

$\displaystyle \sum_{k=1}^n f(1+\frac{3k}{n})(\frac{3}{n})$

$\displaystyle = \frac{3}{n}\sum_{k=1}^n (\frac{3k}{n}+3) = \frac{3}{n}\sum_{k=1}^n \frac{3k}{n}\ +\ \frac{3}{n}\sum_{k=1}^n 3$

$\displaystyle = \frac{9}{n^2}\sum_{k=1}^n k\ +\ 9 = \frac{9(n+1)}{2n} + 9$

now, what we actually want is: $\displaystyle \lim_{n \to \infty} (\frac{9(n+1)}{2n} + 9)$

i've done the hard part, now you do the easy part (and convince yourself that this matches your intuitive idea of the area of the region bounded by the x-axis, the lines x = 1 and x = 4, and the graph of f).

7. Re: Using limits to evaluate integral?

I think I understand know I'll have to try and solve it.

But what confuses me is when I have (4/n)(4i/n+2) Would I just use distributive property

Or add it first if I add it first I would need like terms

And If I did distributive property I dont know if there is a sum for 8/n

8. Re: Using limits to evaluate integral?

i'm not sure why you keep fixating on 4. your interval ends at 4, but it starts at 1, so it has length 4-1 = 3. if you chop an interval 3 units long into n pieces, each piece has length 3/n, not 4/n.

also f(xi) is NOT 4xi/n + 2. the i-th interval (out of our n) goes from 1 + (4-1)(i-1)/n to 1 + (4-1)(i)/n.

if we pick the right-hand endpoint of this for xi, then xi = 1 + 3i/n (i used k instead of i in my earlier post, its a personal preference).

f(xi) = (xi) + 2 = (1 + 3i/n) + 2 = 3i/n + 3. look, no 4's. now if we sum:

$\displaystyle \sum_{i=1}^n (3i/n + 3)(3/n)$, we can take the factor 3/n out of the sum (by the distributive law).

that leaves us with:

$\displaystyle (3/n)[\sum_{i=1}^n\ i/n + \sum_{i=1}^n 3]$

the second summation is just n 3's, so we get:

$\displaystyle (3/n)[\sum_{i=1}^n\ i/n +\ 3] = (3/n)\sum_{i=1}^n\ i/n + (3/n)(3n)$

in the second sum, the n's cancel, leaving (3)(3) = 9. so our sum is 9 plus something, the something being:

$\displaystyle (3/n)\sum_{i=1}^n\ i/n$. again we can take the factor of 1/n out in front, so we have:

$\displaystyle [(3/n^2)\sum_{i=1}^n\ i\] +\ 9$

now surely you know that $\displaystyle \sum_{i=1}^n\ i = n(n+1)/2$,

so our formula becomes $\displaystyle (3/n^2)(n(n+1))/2 + 9 = (3/n)((n+1)/2) + 9$,

which is the same as (3/2)((n+1)/n) + 9, as in my previous post.

i understand that calculating riemann sums is a tedious process, which requires careful attention to the algebra, but there is no easy way around it.

9. Re: Using limits to evaluate integral?

Hmm your explanation is crisp and clear like 7 up. But I wonder how did you get +3 into (3/n)(3n).

10. Re: Using limits to evaluate integral?

Originally Posted by homeylova223
Hmm your explanation is crisp and clear like 7 up. But I wonder how did you get +3 into (3/n)(3n).
Your question makes no sense. (3/n)(3n) is equal to 9, which is what Deveno said!

If you are wondering where the (3n) came from in (3/n)(3n), you're expected to know the basic property that $\displaystyle \sum_{i=1}^n 3 = 3n$.