5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 271

Example 4

Using a calculator, find the area bounded byf(x)=x^3 +x^2 − 6 xand thex-axis. See Figure
12.3-6.

[−4,3] by [−6,10]

Figure 12.3-6

Step 1. Entery 1 =x∧ 3 +x∧ 2 − 6 x.

Step 2. Enter

∫

(abs(x∧ 3 +x∧ 2 − 6 ∗x),x,−3, 2)and obtain 21.083.

Example 5

The area under the curvey=exfromx=0tox=kis 1. Find the value ofk.

Area=

∫k

0

exdx=ex]k 0 =ek−e^0 =ek− 1 ⇒ek=2. Take ln of both sides:

ln(ek)=ln 2;k=ln 2.

Example 6

The region bounded by thex-axis and the graph ofy=sinxbetweenx=0 andx=πis
divided into 2 regions by the linex=k. If the area of the region for 0≤x≤kis twice the
area of the regionk≤x≤π, findk. (See Figure 12.3-7.)

1

0 k

y = sin x

y

x

Figure 12.3-7