Write a Riemann sum and then a definite integral representing the area of the region

1. Feb 22, 2009

Mgeorges

1. The problem statement, all variables and given/known data

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown in the figure below where the upper line is defined by 6x + y = 12 and the other line is defined by y=x^2-4. The figure, which I cant get on here, is just the area bounded between those two equations. I do not need the Riemann sum, I just need to find: (a) What is the approximate area of the strip with respect to x (the strip is horizontal)? I found the limits of integration which is from [0,2]
2. Relevant equations

3. The attempt at a solution
I have no idea how to find the area of the strip, and after that I can figure out the integral with no problem.

2. Feb 23, 2009

Dick

Re: Write a Riemann sum and then a definite integral representing the area of the reg

If you can find the integral with no problems, how come you need to find out the area of the strip first? And those two curves intersect at x=2 but not x=0. Did the figure you didn't show only tell you to consider x>=0?

3. Feb 23, 2009

Mgeorges

Re: Write a Riemann sum and then a definite integral representing the area of the reg

I have to find the area of the strip, and from there I can get the integral..The method were using is volumes by slicing, and the [0,2] were the bounds for the integral. I do not know how to set up this definite integral anyway.

4. Feb 23, 2009

Dick

Re: Write a Riemann sum and then a definite integral representing the area of the reg

To get a Riemann sum you create a partition of [0,2] and then calculate an upper sum or a lower sum to approximate the area of the region. It's just a sum of rectangle areas. If you want to set up an integral you have to figure out which curve is above the other in the x interval [0,2], subtract the lower value from the upper value and integrate over [0,2]. That's the exact area.

5. Feb 23, 2009

HallsofIvy

Staff Emeritus
Re: Write a Riemann sum and then a definite integral representing the area of the reg

The rectangle making part of the Riemann sum has base $\Delta x$ and height equal to the distance between the two curves for some value of x in the interval. Since y= 12- 6x is always above y= x2- 4, that distance is (12- 6x)- (x2- 4)= 16- 6x- x2. The area is the product of those two.

You say the integral is from 0 to 2. I will repeat Dick's question: are you told that, separately in the question? In your first post, you only said that you were finding the area between the graphs: and that runs from x= -8 to x= 2.