Find the area of a region between two curves using integration
Find the area of a region between intersecting curves using integration
Describe integration as an accumulation process
With just small modifications you can extend the idea of definite integrals (area under a curve and above (or below) the x-axis) to area between two curves. Consider f(x) and g(x) that are continuous on an interval [a, b] where the graph of g lies below that of graph of f
This representative rectangle is a HUGE idea. It is the basis for your entire setup, so hopefully it makes sense to you. Right now it seems trivial, but if you get into the habit of sketching in a representative rectangle, volumes of revolution (7.2 and 7.3) will make sense. Area = height ∙ width We’re just adding up areas again!
As you can see from the graphs, it doesn’t really matter where the two graphs are on the plane, as long as f(x) is ABOVE g(x) for the interval [a, b] Representative rectangles are used throughout this chapter. A vertical rectangle (of width Δ x) implies integration with respect to x. A horizontal rectangle (of width Δ y) implies integration with respect to y.
Ex 1 p447 Finding the Area of a Region Between Two Curves Find the area of the region bounded by the graphs of dx: adding up rectangles from left to right of width Δ x Left-most value of x Right-most value of x Top function Bottom function
In the last example, values for a and b have to be given explicitly. A more common problem occurs if the graphs intersect and you must calculate the values for a and b . Ex 2 p. 448 A Region Lying Between Two Intersecting Graphs. Solution: Graph both and find intersection points. The x-values will be the interval we are putting for lower and upper limits of integration. Draw in representative rectangle. (1, 0) (2, 1) Then set up integral:
Ex 3 p. 448 A Region Lying Between Two Intersecting Graphs Since the sine and cosine curves intersect infinitely many times, you are to find only one such region. Find the region bound by f(x) = sin x and g(x) = cos x I decided to look at region from x = -3 π /4 and x = π /4 What function is on top in this interval? If you are solving by calculator, make sure you realize that you are looking for f(x) – g(x)! Not just area under the last curve you graphed, which might be just g(x) or just f(x).
If two curves intersect at more than two points, then you must split up into more than one interval based on the x-values of ALL the points of intersection. Make sure you realize which graph is on top in a particular interval. Ex 4 p. 449 Curves that intersect at more than two points. Find the area of the region between the graphs of f(x) = 3x 3 – x 2 – 10x and g(x) = -x 2 + 2x f(x) = g(x) to find intersections: set = 0, then solve. x = 0, 2. Which function is on top for [-2, 0]? Which is on top for [0, 2]? Be careful with algebra here! Since these are complicated enough, I will let you find the definite integral by calculator. However, you must show what integrals in simplest form you are entering with correct limits. It is a coincidence that both regions are the same area!
You would have gotten the wrong result if you had merely integrated f(x)-g(x) from [-2, 2] Ex 5 p.450 Horizontal Representative Rectangles Find the area of the region bounded by the graphs of x = 3 – y 2 and x = y + 1. (Note: these could also have been written f(y) = 3 – y 2 and g(y) = y + 1) You have a couple of choices when trying to visualize them: Graph by hand, or graph the inverse functions by calculator, turn them off, and then go to DRAW menu, choose DrawInv and first select VARS FUNCTION Y 1 and then repeat for Y 2 Notice that the representative rectangle is horizontal, so the widths will by ∆y and the heights will be the difference between the function further to the right and the one further to the left! You will also need to find points of intersection: Set x=x or in other words, 3 – y 2 = y + 1 and solve for y. This leads to (y+2)(y-1)=0, and y = -2, 1 Lower y Upper y Right-most Left-most Notice that we just worked with one integral here!
With this particular graph, we can choose to integrate with respect to y, like we just did, or with respect to x, as shown here or on p.450. Vertical rectangles with respect to x is MUCH more of a pain! With some setups, it is impossible to even do this. So get your mind used to seeing the problems as a bunch of vertical rectangles, or as a bunch of horizontal rectangles, whichever gives you easier/possible calculations. For example, how would you even do a problem like finding the area between f(y) = y 3 -3y and g(y) = y from y = 0 to y = 2 as vertical rectangles?