276 STEP 4. Review the Knowledge You Need to Score High
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: iff′>0, then f is increasing, and if f′′>0 then the graph of f is
concave upward.
12.4 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 12.4-1.)
ab
y
0 x
Figure 12.4-1
Note: A cross section of a solid is perpendicular to the height of the solid.