Finding the Area Between Curves

1. Finding the Area Between Curves Application of Integration

2. Notes to BC students: • I hope everyone had great holidays, I did, including experiencing a blizzard, but now I’m sick… • Since we missed the time before the holidays, some Unit 6 topic(s) will be moved to Quarter III. • This applies to both morning and afternoon classes.

3. The problem is to find the area between two curves, so we start with a couple of friendly calculus curves. The first is , or .

4. And the second is

5. A closer look:

6. We are interested in finding the area of the purple region.

7. Let h be the distance between the two curves. h

8. Notice how h changes as we move from left to right. h

9. Since h is the distance from the upper to lower curve. This is simply the difference of the two y-coordinates. This means that

10. We can find the total area between the curves by integrating h between the points of intersection.

11. Note that the two curves intersect at the origin and at (1,1).

12. The area between the curves is The 0 and 1 are the starting and ending values of x.

13. Further, The area is

14. We can evaluate the integral using the Fundamental Theorem of the Calculus.

15. As a second example, find the area between First, we need to graph the functions and see the defined area.

16. f g

17. Zooming in: g f Notice that the upper intersection is not made of simple values.

18. Later, we will find the intersection. First, we define h. g f Notice that h is the difference between the two x-coordinates. h

19. Notice this distance uses coordinates from the right function minus coordinates from the left function. To have distance be a positive number one must always subtract a smaller from a larger one.

20. As with the first example we integrate h from beginning to end. We see that the origin is one point of intersection. We need to find the other point of intersection.

21. Finally, the area is This is a good time to use your calculator! Note that in this example the limits of integration are y-values, and the integrand is a function of y.

22. There are several points that should be made: • Graph the functions. • Decide whether you will work in vertical or horizontal distances. Use the one that it easiest for the problem. n.b. This is not always x! • Distance is always positive, remember to subtract the smaller value from the larger one, whether using x or y.