section 5.3 - volumes by slicing 7.3 solids of revolution i can use the definite integral to compute...
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Section 5.3 - Volumes by Slicing
7.3Solids of Revolution
I can use the definite integral to compute the volume of certain solids.
Day 2:
1. 02111a-dLet f be the function defined by
a. Write an equation of the line normal to the graph of f at x = 1.
b. For what values of x is the derivative of f, f ‘ (x), not continuous? Justify your answer.
c. Determine the limit of the derivative at each point of discontinuity found in part (b).
d. Can be completed using the method of u-substitution? If yes, complete the integration. If no, explain why u-substitution
cannot be used for
2
4 8( )
5 14
xf x
x x
( )f x dx( )f x dx
rotating region
The development of the volume created by rotating region
between the x-axis and the curve y = x, with x
between 1 and 4 about the y-axis.
The development of the volume created by the region between
the x-axis, x = 0, x = 2 , and the curve y 2 sinx
is revolved about the x-axis
2
The development of the volume created by the region between
2the x-axis and the curve y is revolved
1 x 2
about the x-axis.
rotating region
rotating region
Find the volume of the solid generated by revolving the regionsabout the x-axis.2y x x and y 0 bounded by
12
0
r dx 1
22
0
dx xx
Find the volume of the solid generated by revolving the regionsabout the x-axis.2y 3x x and y 0 bounded by
02
3
dxr
20
2
3
dx3x x
Find the volume of the solid generated by revolving the regionsabout the y-axis.1
2y x, x 0, y 2 bounded by
22
0
r dy 2
2
0
d2y y
Find the volume of the solid generated by revolving the regionsabout the x-axis.2y 2sin2x, 0 x bounded by
/ 22
0
xr d
/ 2
2
0
dx2sin2x
Find the volume of the solid generated by revolving the regionsabout the line y = -1.2y 3 x , y 1 bounded by
22
2
dxr
2
22
2
3 x 1 dx
Let R be the first quadrant region enclosed by the graph ofxy 2e and x k
a) Find the area of R in terms of k.
b) Find the volume of the solid generated when R is rotated about the x-axis in terms of k.
c) What is the volume in part (b) as k approaches infinity?
NO CALCULATOR
Let R be the first quadrant region enclosed by the graph ofxy 2e and x k
a) Find the area of R in terms of k.
kx x k k
0
0
2e dx 2e | 2e 2
Let R be the first quadrant region enclosed by the graph ofxy 2e and x k
b) Find the volume of the solid generated when R is rotated about the x-axis in terms of k.
k
2x
0
2e dx
k
2x
0
4 e dxu 2x
du 2dx
uk
0
e d2 u 2x k
02 e | 2k2 e 2
Let R be the first quadrant region enclosed by the graph ofxy 2e and x k
c) What is the volume in part (b) as k approaches infinity?
2k
klim 2 e 2 2
Let R be the region in the first quadrant under the graph of
3
8y for 1, 8
x
a) Find the area of R.
b) The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k?
c) Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.
CALCULATOR REQUIRED
Let R be the region in the first quadrant under the graph of
3
8y for 1, 8
x
a) Find the area of R.
8
31
8dx 36
x
Let R be the region in the first quadrant under the graph of
3
8y for 1, 8
x
b) The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k?
k
31
8dx
x
A
536
12
2/ 3 k112x | 15
2/ 32k 11 12 5
k 3.375
Let R be the region in the first quadrant under the graph of
3
8y for 1, 8
x
c) Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.
Cross Sections
28
31
8dx 192
x
The base of a solid is the circle . Each section of thesolid cut by a plane perpendicular to the x-axis is a square withone edge in the base of the solid. Find the volume of thesolid in terms of a.
2 2 2x y a
a 2
2 2
a
2 a x dx
a
2 2
a
4a 4x dx
32 a
a
4x4a x |
3
316a
3
Let R be the region in the first quadrant that is enclosed by the
graph of f x ln x 1 , the x-axis and the line x = e. What is
the volume of the solid generated when R is rotated about the
line y = -1?
A.
5.037 B. 6.545 C. 10.073 D. 20.146 E. 28.686
CALCULATOR REQUIRED
e
2 2
0
ln x 1 1 1 dx 20.14627352 D
Let R be the region marked in the first quadrant enclosed bythe y-axis and the graphs of as shown in the figure below
2y 4 x and y 1 2sinx
R
a) Setup but do not evaluate the integral representing the volume of the solid generated when R is revolved around the x-axis.
b) Setup, but do not evaluate the integral representing the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares.
1.102
2 22
0
4 x 1 2sinx dx
1.102
22
0
4 x 1 2sinx dx
Let R be the region in the first quadrant bounded above by thegraph of f(x) = 3 cos x and below by the graph of 2xg x e
a) Setup, but do not evaluate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the x-axis.
b) Let the base of a solid be the region R. If all cross sections perpendicular to the x-axis are equilateral triangles, setup, but do not evaluate, an integral expression of a single variable for the volume of the solid.
20.836 22 x
0
3cosx e dx
20.836 2
x
0
33cosx e dx
4
The volume of the solid generated by revolving the first quadrantregion bounded by the curve and the lines x = ln 3 andy = 1 about the x-axis is closest to
x / 2y e
a) 2.79 b) 2.82 c) 2.85 d) 2.88 e) 2.91
ln3
2 2x / 2
0
e 1 dx
The base of a solid is a right triangle whose perpendicular sideshave lengths 6 and 4. Each plane section of the solidperpendicular to the side of length 6 is a semicircle whosediameter lies in the plane of the triangle. The volume of the solidin cubic units is: a) 2pi b) 4pi c) 8pi d) 16pi e) 24pi
26
0
1 1dx
2
2x
2 3
2
322
0
222
0
24 2
0
2 4
The volume of the solid generated by rotating about the x-axis
the region enclosed between the curve y 3x and the line
y 6x is given by
A. 6x 3x dx
B. 6x 3x dx
C. 9x 36x dx
D. 36x 9x d
2
0
22
0
x
E. 6x 3x dx
CALCULATOR REQUIRED
NO CALCULATORThe base of a solid is the region in the first quadrant bounded by
the curve y sinx for 0 x . If each cross section of the
solid perpendicular to the x-axis is a square, the volume of the
solid, in cu
bic units, is:
A. 0 B. 1 C. 2 D. 3 E. 4
2
0
0 0
sinx dx sinxdx cos x | 1 1 2 C
Let R be the region in the first quadrant above by the graph of
f x 2Arc tanx and below by the graph of y = x. What is the volume
of the solid generated when R is rotated about the x-axis?
A. 1.21
B. 2.28 C. 2.69 D. 6.66 E. 7.15
CALCULATOR REQUIRED
NO CALCULATORThe base of a solid is a right triangle whose perpendicular sides
have lengths 6 and 4. Each plane section of the solid perpendicular
to the side of length 6 is a semicircle whose diameter lies in the
plane of the triangle. The volume, in cubic units, of the solid is:
A. 2 B. 4 C. 8 D. 16 E. 24 2
6 62
0 0
3 60
2x1 13 dx x dx
2 2 18
x |544 B
x / 2
The volume of the solid generated by revolving the first quadrant
region bounded by the curve y e and the lines x = ln 3 and y = 1
about the x-axis is:
A. 2.80 B. 2.83 C. 2.86
D. 2.89 E. 2.92
CALCULATOR REQUIRED
CALCULATOR REQUIRED
2 2
The region S is represented by the area between the graphs of
f x 0.5x 2x 4 and g x 2 4 4x x . Write, but do
not evaluate, a definite integral which represents:
a. the volume of a solid with base S if eac
h cross section of
the solid perpendicular to the x-axis is a semi-circle.
b. the volume generated by rotating region S around the line
y = 5.
24
0
g x f xdx
2 2
4
2 2
0
5 f x 5 g x dx
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