AP Calculus Chapter 6 Applications of the Integral

AP Calculus - Chapter 6 Applications of the Integral 6. 1: Area Between Graphs Find the area between the graphs of two functions by partitioning the x-axis Find the area between the graphs of two functions by partitioning the y-axis

This lesson is about determining the area between two functions, as opposed to determining the area between a function and the x axis. Looking back at Riemann sums, how did we determine how wide the rectangles were? The idea is that we take the larger quantity minus the smaller quantity and we will find the distance between the two numbers. The same thing can be applied to areas under curves. If one curve is above another curve, and we are interested in finding the area between the two curves, we can subtract the areas.

Ignoring the rectangle, let's look at how we could find the area between these two curves by using integrals. Ex: Find the area between the curves above.

An important note: The formula from the previous slide will always be true, regardless of whether the curves are above the x-axis or below, so long as f(x) is always greater than g(x) on the interval given. Ex: Find the area A of the region enclosed by the graphs below. This problem does not specify x values. It just tells you to find the area of the "enclosed region". To find the enclosed region, we need to figure out. . . Where the functions intersect Which function is above and which is below.

Ex: Find the area of the region enclosed by the graphs of y = sin x and y = cos x from the y-axis to their first point of intersection.

Find the area A of the region enclosed by the graph of f(x), the x-axis, and the lines x = 0 and x = 5

y-axis partitioning: Sometimes it is easier to use the y-axis to determine the area instead of the x-axis. Let's look at an example. This region is difficult to find because the top and bottom borders are not defined by the curves or the x-axis, but if we were to use the y-axis we could find the region easily.

Find the area A enclosed by the graphs.

Find the area A of the region enclosed by the graphs below and the x-axis.