Calculating the Area of a Strip Using a Riemann Sum

[Mgeorges]

Homework Statement

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 can't 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]

Homework Equations

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.

 

[Dick]

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?

 

[Mgeorges]

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.

 

[Dick]

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.

 

[HallsofIvy]

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.