5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 285

x

y

y = –3

y = ex

0

–3

ln 2

Figure 12.4-11

Step 2. Determine the radius from a cross section.

r=y−(−3)=y+ 3 =ex+ 3

Step 3. Set up an integral.

V=π

∫ln2

0

(ex+ 3 )^2 dx

Step 4. Evaluate the integral.

Enter

∫ (

π(e∧(x)+ 3 )∧2, x, 0 ln (2)

)

and obtainπ

(

9ln2+

15

2

)

= 13. 7383 π.

The volume of the solid is approximately 13. 7383 π.

TIP • Remember: iff′is increasing, thenf′′>0 and the graph offis concave upward.

The Washer Method

The volume of a solid (with a hole in the middle) generated by revolving a region bounded

by 2 curves:

About thex-axis:

V=π

∫b

a

[

(f(x))^2 −(g(x))^2

]

dx; wheref(x)=outer radius andg(x)=inner radius.

About they-axis:

V=π

∫d

c

[

(p(y))^2 −(q(y))^2

]

dy; wherep(y)=outer radius andq(y)=inner radius.