Question

Help please! (: Integral problem using Riemann sum. I Could not figure out how to start it.


Problem attached below!(:

Answer 1

what textbook is this?

Answer 2

it's not a text book. but it's calc 1. @mathbrz

Answer 3

@pdd21 oh man, I remember doing this, but I don't exactly remember the steps to prove this.

Answer 4

Yeah I have those notes, but I have no idea how to do this problem. :/

Answer 5

I think this problem is asking you to use the definition of a definite integral to evaluate that

Answer 6

Here are some examples of doing just that:

http://www.dawsoncollege.qc.ca/public/72b18975-8251-444e-8af8-224b7df11fb7/programs/disciplines/math/coursesupplements/supplementary_notes_-_riemann_sum.pdf

Answer 7

I still don't get it :/

Answer 8

Here is a picture of f for p=q=b=1 and n=5

Answer 9

If you look at the upper rectangle, what you lose from under the curve, you gain from the part of the rectangle outside. That is why, you get the exact area under the curve. Which 2 in this case

Answer 10

Here is the picture for n=8

Answer 11

so exact area is 2? how do I explain it in words rather then a graph for (a)?

Answer 12

@eliassaab

Answer 13

You can compute the area of each rectangles in terms of p, q and b and add them up amd you get the same answer as \[\int_0^b f(x) d x\]

Answer 14

how would you do that? @eliassaab

Answer 15

like how would I write it out in integral form? @eliassaab

Answer 16

You can show that the sum of the area of the rectangles is
\[
\frac{b (b (n-1) p+b p+n q)}{n}=b (b p+q)=\int_0^b f(x) \, dx
\]

Answer 17

The area of the first rectangle from the left is
\[
\frac{b \left(\frac{b p}{n}+q\right)}{n}
\]

Answer 18

The second rectangle is
\[
\frac{b \left(\frac{3 b p}{n}+q\right)}{n}
\]
and so fourth

Answer 19

The last rectangle's area is
\[
\frac{b \left(2 p \left(b-\frac{b}{2 n}\right)+q\right)}{n}
\]

Answer 20

Add them up and you are done.

Answer 21

I like the geometric (graph) proof better.

Answer 22

how do you explain the importance of riemann sum to finding the exact area? @eliassaab

Answer 23

It only find the exact area when you have a line.

Answer 24

When you get to do the Simpson's rule, it will get the exact area for polynomials of degree 3 or less