Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

Area Bounded by Two Curves: See Figure 12.3-8.

### Example 1

Find the area of the region bounded by the graphs of *f* (*x*)=*x*^{3} and *g*(*x* )=*x*. (See Figure 12.3-9.)

Step 1. Sketch the graphs of *f* (*x* ) and *g* (*x* ).

Step 2. Find the points of intersection.

Step 3. Set up integrals.

Note: You can use the symmetry of the graphs and let area

Analternate solution is to find the area using a calculator. Enter and obtain .

### Example 2

Find the area of the region bounded by the curve *y* =*e*^{x}, the *y*-axis and the line *y* =*e*^{2}.

Step 1. Sketch a graph. See Figure 12.3-10.

Step 2. Find the point of intersection. Set *e*^{2} =*e*^{x} *x* =2.

Step 3. Set up an integral:

Or using a calculator, enter and obtain (*e*^{2} +1).

### Example 3

Using a calculator, find the area of the region bounded by *y* = sin *x* and between 0≤ *x* ≤ π.

Step 1. Sketch a graph. See Figure 12.3-11.

Step 2. Find the points of intersection.

- Using the [

*Intersection*] function of the calculator, the intersection points are

*x*=0 and

*x*=1.89549.

Step 3. Enter nInt(sin(*x* ) &8211; .5*x*, *x*, 0, 1.89549) and obtain 0.420798 ≈ 0.421.

- (Note: You could also use the function on your calculator and get the same result.)

### Example 4

Find the area of the region bounded by the curve *x y* =1 and the lines *y* = –5, *x* =e, and *x* =*e*^{3}.

Step 1. Sketch a graph. See Figure 12.3-12.

Step 2. Set up an integral.

Step 3. Evaluate the integral.

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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