The figure shows graphs of the marginal revenue function $ R' $ and the marginal cost function $ C' $ for a manufacturer. [Recall from Section 4.7 that $ R(x) $ and $ C(x) $ represent the revenue and cost when $ x $ units are manufactured. Assume that $ R $ and $ C $ are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.
Applications of Integration
this problem from your textbook is really great because this gets to the heart of the definition of what an integral is. And this problem we're using our knowledge of the midpoint and the area under a curve defined the area essentially of a shaded region. And now why this relates to the definition of an integral is because we're finding essentially an infinite number of rectangles under this curve and adding all the areas together to get the overall area. So the first thing that we can do is find the midpoint of all of these intervals. So the first intervals is 50 60. And then if you took the average of those two numbers, you get 55 so that would be our midpoint. Then we do it for the coordinate 60 70 are midpoint would be 65 and for 78 midpoint would be 75 then from 80 90 are midpoint would be 85 then from 90 to 100 are midpoint would be 95. So then, in this problem, we're dealing with an application about costing revenue. So we know that our profit p prime of X would be equivalent to our revenue, Our prime of X minus our cost see prime of X And then what we find is the definition of Delta X. Remember, we're looking at thes triangles, so Delta X would be B minus A over n and X would be from 50 to 100 so that would be be would be 108 would be 50 and then end is asking What is the width of these triangles? Well, it's five, because that's the difference between each of these mid points. So then we can find Delta X, remember, its be minus a over n so Delta X would be 100 minus 50/5 and we would get 10. But now remember, we have to find the area of the shaded region, which I've just noted as a letter. A is also area. It's just the shaded region under the curve. So the way we would do that is we will take Delta X and multiply that by this entire parentheses. E Now note. It's very important to keep track of our brackets here. So we have our prime of 55 minus C prime of 55 plus our prime of 65 minus C prime of 65 plus our prime of 75 minus C prime of 75 plus our prime of 85 minus C, prime of 85 and then plus our prime of 95 minus C prime of 95. Now, if you're confused, How I got that All I did was I plugged in the mid point for each of these intervals into our equation for P prime of X. Now you can look at the graph in your textbook to find these values. So we would take Delta X times, remember, keep track of our brackets 2.4 minus 0.85 plus 2.2 minus 0.9 plus to minus one plus 1.8 minus 1.1 plus 1.7 minus 1.2. Then remember, we found Delta X s 10 and we simplified all of this. We would get 5.3. We multiply that out and we would find the area of the shaded region of our curve is 55 53. Pardon me. So I hope that this problem helped you understand how we can use the knowledge of midpoint and Delta X, or the really the essential definition of an integral by finding the area of an infinite number of rectangles under a curve to find the area.