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

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

Areas and Volumes 289

Step 2: Set up an integral.

Area =

∫e 3

e

(

1

x

−(−5)

)

dx.

Step 3: Evaluate the integral.

Area =

∫e 3

e

(

1

x

−(−5)

)

dx=

∫e 3

e

(

1

x

+ 5

)

dx

=ln|x|+ 5 x]e

3

e =

[

ln(e^3 )+5(e^3 )

]

−[ln(e)+5(e)]

= 3 + 5 e^3 − 1 − 5 e= 2 − 5 e+ 5 e^3.

TIP • Remember: If f′>0, then f is increasing, and if f′′>0, then the graph of f is

concave upward.

13.4 Volumes and Definite Integrals

Main Concepts:Solids with Known Cross Sections, The Disc Method,

The Washer Method

Solids with Known Cross Sections

IfA(x) is the area of a cross section of a solid andA(x) is continuous on [a,b], then the

volume of the solid fromx=atox=bis:

V=

∫b

a

A(x)dx.

(See Figure 13.4-1.)

ab

y

x

0

Figure 13.4-1

Note: A cross section of a solid is perpendicular to the height of the solid.