Finding the Area Between Curves Application of Integration

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.

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

• And the second is

• A closer look:

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

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

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

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

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

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

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

Further, The area is

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

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

f g

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

Later, we will find the intersection. First, we define h. g Notice that h is the difference between the two xcoordinates. f h

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.

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.

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.

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.