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Integral of x^2 - 4x + 2, from 1 to 3, using Riemann sums

mackin101

New member
Joined
Sep 29, 2017
Messages
1
Hi! My question is:
Find the integral of x[SUP]2[/SUP]-4x+2, with the upper limit being 3 and the lower limit being 1 using Riemann sums with general n and right end point.

Obviously, I have no clue how to input this properly, so if you could help with that too, it would be appreciated. I can do the right end point without the exponent, however I don't know where to start on this one. I would be grateful if you would even give me a different example with a similar format.

Thanks in advance!
 

JeffM

Elite Member
Joined
Sep 14, 2012
Messages
3,174
Hi! My question is:
Find the integral of x[SUP]2[/SUP]-4x+2, with the upper limit being 3 and the lower limit being 1 using Riemann sums with general n and right end point.

Obviously, I have no clue how to input this properly, so if you could help with that too, it would be appreciated. I can do the right end point without the exponent, however I don't know where to start on this one. I would be grateful if you would even give me a different example with a similar format.

Thanks in advance!
The question as you have asked it is unclear: "general n"?

I am guessing that they want you to start by setting up the general formula for a Riemann sum for this function over the specified interval.

If we have n rectangles of equal width over the interval [3, 1], what is the width of each rectangle?

\(\displaystyle \dfrac{3 - 1}{n} = \dfrac{2}{n}.\)

That's easy. Next thing to work out is almost as easy. You are to do this to the right so where on the x-axis is the right vertical of the jth rectangle?

\(\displaystyle x_j = 1 + j * \dfrac{2}{n}.\) Do you see why?

The first right-hand vertical is 1 width to the right of 1. The second right-hand vertical is 2 widths to the right of 1, correct. Alright, the next thing to do may involve some messy algebra, but is conceptually simple. What is the height of the jth rectangle?

\(\displaystyle h_j = f(x_j) = WHAT?\)

Now you can compute the area of the jth rectangle.

\(\displaystyle a_j = f(x_j) * \dfrac{2}{n} = WHAT?\)

Let's see where you are before going on.
 

stapel

Super Moderator
Staff member
Joined
Feb 4, 2004
Messages
15,937
Find the integral of x[SUP]2[/SUP]-4x+2, with the upper limit being 3 and the lower limit being 1 using Riemann sums with general n and right end point.

Obviously, I have no clue how to input this properly...

Why "obviously"? What do you mean by "inputting"? Are you saying that you're unsure how to apply the formulaic method they gave you in the book? If so, please reply showing what you've tried, so we can see where you're needing help. If not, please reply with clarification.

I can do the right end point without the exponent, however I don't know where to start on this one.
What difference are you seeing, between applying the formula with a linear integrand and applying it with a higher-degree integrand?

Please be specific. Thank you! ;)
 
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