5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 287

Note that you can use the symmetry of the graphs and let area = 2

∫ 1

0

(

x−x^3

)

dx.

An alternate solution is to find the area using a calculator:

Enter

∫

(abs(x∧ 3 −x),x,−1, 1) and obtain

1

2

.

Example 2

Find the area of the region bounded by the curvey=ex, they-axis, and the liney=e^2.

Step 1: Sketch a graph. (See Figure 13.3-10.)

y = ex

y = e^2

x

y

1

0 12

Figure 13.3-10

Step 2: Find the point of intersection. Sete^2 =ex⇒x=2.

Step 3: Set up an integral:

Area =

∫ 2

0

(e^2 −ex)dx=(e^2 )x−ex]^20

=(2e^2 −e^2 )−(0−e^0 )

=e^2 + 1.

Or using a calculator, enter

∫

((e∧ 2 −e∧x),x,0,2)and obtain (e^2 +1).