Name: _______________
Calculus Review
Area and Volume F15
1.
Sketch a graph of f ( x) = 3x3 − x2 −10 x and g ( x) = − x 2 + 2 x .
Label the axis, curves, and points of intersection. Shade the region
between the curves and find the area of the shaded region.
For 2 – 4, R is the region bounded by the x-axis and f ( x) = sin x .
2.
Sketch and label R for 0 ≤ x ≤ π.
3.
Find the volume of the solid formed by revolving R about the xaxis.
4.
(Calc) Find the volume of the solid formed by revolving R about
the line y = 2.
For 5 - 9, R is the region between f ( x) = x and g ( x) = x 2 .
5.
Sketch and label the graph of R and shade the area between the
curves. Find the shaded area.
6.
Find the volume formed by rotating R about the x-axis.
7.
Find the volume formed by rotating R about the line y = 3.
8.
Find the volume formed by rotating R about the y-axis.
9.
Find the volume formed by rotating R about the line x = -2.
For 10 – 14, the base of a solid is the area between f ( x) = x + 1 and
g ( x) = x 2 − 1.
10.
Draw and label a sketch of the base.
11.
Find the volume of the solid if the cross sections perpendicular to
the x-axis are squares.
12.
Find the volume of the solid if the cross sections perpendicular to
the x-axis are rectangles with a height of 1.
13.
Find the volume of the solid if the cross sections perpendicular to
the x-axis are equilateral triangles.
14.
Find the volume of the solid if the cross sections perpendicular to
the x-axis are semicircles.
15.
Let R be the area between the curves y = x 2 and y = 4 x − x 2 .
A. Sketch and shade R labeling everything.
B.
Find the area of R.
C.
Find the volume when R is rotated about the line x = -2.
D.
Find the volume when R is rotated about the line y = 6.
E. Find the volume of the solid that has R as it’s base and whose cross
sections are rectangles perpendicular to the x-axis with the base twice as
long as the height.
16. Let R be the area between the curves x = y 2 and x = y + 2 .
A.
Sketch and shade R labeling everything.
B.
Find the area of R.
C.
Find the volume when R is rotated about the y-axis.
D.
Find the volume when R is rotated about the line x = 4.
E.
Find the volume when R is rotated about the line x = 7.
F.
Find the volume of the solid that has R as its base and whose cross
sections are semi-circles perpendicular to the y-axis.
17. Let R be the region between the curves y = e x and y = ln( x + 1) + 2 .
A. Find the area of R.
B. Find the volume of figure formed by rotating region R about the x axis.
18. The rate at which your home consumes electricity is measured in kilowatts. If your home
consumes electricity at the rate of 1 kilowatt for 1 hour, you will be charged for 1 "kilowatthour" of electricity. Suppose that the average consumption rate for a certain home is modeled by
⎛ πt ⎞
the function C (t ) = 4 − 2.4sin ⎜ ⎟ , where C(t) is measured in kilowatts and t is the number of
⎝ 12 ⎠
hours past midnight. Find the average daily consumption for this home, measured in kilowatthours.
(A)
114.3
(B) 86.8
(C) 105.2
(D) 96
(E) None of the Above
19. The velocity in m/sec of a particle moving along the x-axis is given by the function
v(t ) = 22cos 2 (t )sin(t ) , 0 ≤ t ≤ 2π. Find the particle's displacement for the given time interval. If
s(0) = 22, what is the particle's final position at the end of the given time interval?
(A)
0; 22
(B) -22; 0
(C) 2; 24
(D) 22; 44
(E) None of the Above
7t
20. Let v(t ) =
be the velocity, in m/s, of a particle traveling along a line. Find the
1+ t2
total distance traveled by the particle for 0 ≤ t ≤ 3 .
7
(A) 7 ln(9)
(B) 7 ln(10)
(C)
ln(9)
2
7
(D)
(E) None of the above
ln(10)
2
21. (calc ok) Region T is bounded in the first quadrant by f ( x) = ln( x + 1) and g ( x) = −2 x + 4 , as shown below. y
f(x)=ln(x+1)
f(x)=-2x+4
2
1
x
1
2
3
(a) Find the area of region T. Explain why you chose this method.
(b) Let the line y = k divide region T into two parts of equal area. Set up an integral expression that can be
used to find the value of k.
Calc OK: