In Section 14.3, you will learn how to find the area between two curves. Suppose you have two functions *f*(*x*) and *g*(*x*). Also assume that the higher curve is *f*(*x*). We are interested in finding the area from a point *x* = *a* to *x* = *b* between the two curves. We can do this by finding the area below f(x) and above the x-axis,

$latex \displaystyle \int\limits_{a}^{b}{f(x),dx}$

and subtracting the area below g(x) and above the x-axis,

$latex \displaystyle \int\limits_{a}^{b}{g(x),dx}$

Alternately, we can subtract the functions first and then find the area,

$latex \displaystyle \int\limits_{a}^{b}{\left[ f(x)-g(x) \right],dx}$

On Monday, the face-to-face class worked several of these types of problems.

To start these problems, graph the different equations to see the region you are finding the area of.

**Problem 1** Find the area between the curves $latex \displaystyle y={{x}^{2}}-30$ and $latex \displaystyle y=10-3x$.

The key to this problem is to first find where the curves intersect by setting the functions equal to each other…these solutions for the limits on the definite integral.

**Problem 2** Find the area between the curves $latex \displaystyle y={{x}^{2}}-18$ and $latex \displaystyle y=x-6$.

The points of intersection may also be found using a graphing calculator.

In the next problem. the region that is enclosed is a little different since the curves cross. When this happens, the region needs to be broken into two parts.

**Problem 3** Find the area of the region enclosed by *x* = 0, *x* = 6, *y* = 5*x* and *y* = 3*x *+ 10.

The two line cross at *x* = 5. On the left side of this point *y* = 3*x* + 10 is higher than *y* = 5*x*. On the right side of *x* = 5, *y* = 5*x* is higher than *y* = 3*x* + 10. This means the order in which the functions are subtracted must change.

**Problem 4** Find the area enclosed by *x* = -2, *x* = 1, *y* = 2*x*, and $latex \displaystyle y={{x}^{2}}-3$.