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Area Between Two Curves for Definite Integrals for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

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.

Area Between Two Curves

Example 1

Find the area of the region bounded by the graphs of f (x)=x3 and g(x )=x. (See Figure 12.3-9.)

Area Between Two Curves

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 =ex, the y-axis and the line y =e2.

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

Area Between Two Curves

Step 2. Find the point of intersection. Set e2 =ex x =2.

Step 3. Set up an integral:

Or using a calculator, enter and obtain (e2 +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.

Area Between Two Curves

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; .5x, 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 =e3.

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

Area Between Two Curves

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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