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

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

Areas and Volumes 303

4. The base of a solid is the region bounded by the linesy=x,x=1, and thex-axis. The
   cross sections are squares perpendicular to thex-axis. Set up an integral to find the
   volume of the solid. Do not evaluate the integral.
   Answer:Area of cross section=x^2.

Volume of solid=

∫ 1

0

x^2 dx.

5. Set up an integral to find the volume of a solid generated by revolving the region
   bounded by the graph ofy=sinx, where 0≤x≤πand thex-axis, about thex-axis.
   Do not evaluate the integral.
   Answer:Volume =π

∫π

0

(sinx)^2 dx.

6. The area under the curve ofy=

1

x

fromx=atox=5 is approximately 0.916 where

1 ≤a<5. Using your calculator, finda.

Answer:

∫ 5

a

1

x

dx=lnx

∣∣ 5

a=ln 5−lna=^0.^916

lna=ln 5− 0. 916 ≈. 693

a≈e^0.^693 ≈ 2

13.6 Practice Problems

Part A The use of a calculator is not

allowed.

1. LetF(x)=

∫x

0

f(t)dtwhere the graph of

f is given in Figure 13.6-1.

01234

4

• 4

5

f

y

x

Figure 13.6-1

(a) EvaluateF(0),F(3), andF(5).

(b) On what interval(s) isFdecreasing?

(c) At what value oftdoesFhave a

maximum value?

(d) On what interval isFconcave upward?

2. Find the area of the region(s) enclosed by
   the curve f(x)=x^3 , thex-axis, and the lines
   x=−1 andx=2.


3. Find the area of the region(s) enclosed by
   the curvey=

∣∣

2 x− 6

∣∣

, thex-axis, and the

linesx=0 andx=4.

4. Find the approximate area under the curve

f(x)=

1

x

fromx=1tox=5, using four

right-endpoint rectangles of equal lengths.

5. Find the approximate area under the curve
   y=x^2 +1 fromx=0tox=3, using the
   Trapezoidal Rule withn=3.