5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 295

12.7 Practice Problems

Part A The use of a calculator is not

allowed.

1. LetF(x)=

∫x

0

f(t)dt, where the graph of

f is given in Figure 12.7-1.

01234

4

–4

5

f

y

x

Figure 12.7-1

(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 up?

2. Find the area of the region(s) enclosed by
   the curvef(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.

6.Find the area of the region bounded by the

graphsy=

√

x,y=−x, andx=4.

7.Find the area of the region bounded by the

curvesx=y^2 andx=4.

8.Find the area of the region bounded by the

graphs of all four equations:

f(x)=sin

(

x

2

)

;x-axis; and the lines,

x=

π

2

andx=π.

9.Find the volume of the solid obtained by

revolving about thex-axis, the region

bounded by the graphs ofy=x^2 +4, the

x-axis, they-axis, and the linesx=3.

10. The area under the curvey=

1

x

fromx= 1

tox=kis 1. Find the value ofk.

11. Find the volume of the solid obtained by
    revolving about they-axis the region
    bounded byx=y^2 +1,x=0,y=−1, and
    y=1.


12. LetRbe the region enclosed by the graph
    y= 3 x, thex-axis, and the linex=4. The
    linex=adivides regionRinto two regions
    such that when the regions are revolved
    about thex-axis, the resulting solids have
    equal volume. Finda.
    Part B Calculators are allowed.


13. Find the volume of the solid obtained by
    revolving about thex-axis the region
    bounded by the graphs of f(x)=x^3 and
    g(x)=x^2.


14. The base of a solid is a region bounded by
    the circlex^2 +y^2 =4. The cross sections of
    the solid perpendicular to thex-axis are
    equilateral triangles. Find the volume of the
    solid.


15. Find the volume of the solid obtained by
    revolving about they-axis, the region
    bounded by the curvesx=y^2 , andy=x−2.