Worksheet 3.3: Introduction to Triple Integrals · BSU Math 275 (Ultman) Worksheet 3.3:...
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BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals From the Toolbox (what you need from previous classes) Calc I/II: Be able to evaluate integrals of a single variable. Calc III: Be able to find limits of integration for and evaluate double integrals over a planar region. Be able to sketch basic surfaces (planes, spheres, cones, paraboloids) and three-dimensional regions bounded by these surfaces. Goals In this worksheet, you will: Set up triple integrals. Identify the regions over which a triple integral is being evaluated. Warm-Up: Cartesian Volume Element dV The volume element dV in Cartesian coordinates is the volume of an infinitesimal box: dV = dx dy dz = dx dz dy = dy dx dz = dy dz dx = dz dx dy = dz dy dx Using the “infinite magnifying glass”, sketch an infinitesimal box whose volume is dV below. Label the sides of the box.
Transcript of Worksheet 3.3: Introduction to Triple Integrals · BSU Math 275 (Ultman) Worksheet 3.3:...
BSU Math 275 (Ultman)
Worksheet 3.3: Introduction to Triple Integrals
From the Toolbox (what you need from previous classes)
� Calc I/II: Be able to evaluate integrals of a single variable.
� Calc III: Be able to find limits of integration for and evaluate double integrals over a planar
region.
� Be able to sketch basic surfaces (planes, spheres, cones, paraboloids) and three-dimensional
regions bounded by these surfaces.
Goals
In this worksheet, you will:
� Set up triple integrals.
� Identify the regions over which a triple integral is being evaluated.
Warm-Up: Cartesian Volume Element dV
The volume element dV in Cartesian coordinates is the volume of an infinitesimal box:
dV = dx dy dz = dx dz dy = dy dx dz = dy dz dx = dz dx dy = dz dy dx
Using the “infinite magnifying glass”, sketch an infinitesimal box whose volume is dV below.
Label the sides of the box.
BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals 1
Model 1: Finding Limits of Integration, Part I
Diagram 1A
W is the 3-dimensional region above the
triangle D (in the xy -plane) and below the
paraboloid z = x2 + y 2.
Diagram 1B
D is a triangular region in the xy -plane.
Begin with Q1-Q3 to find these limits of integration.
˚W
f (x, y , z) dV =
ˆ x=x=
ˆ y=y=
ˆ z=z=
f (x, y , z) dz dy dx
Critical Thinking Questions
In this section, you will find the limits of integration for the 3-d region W given in Model 1.
(Q1) Using the order of integration dV = dz dy dx , the innermost differential is dz .
Go to Diagram 1A, and draw an arrow parallel to the z-axis, starting below the triangular
region D in the xy -plane, and ending above the paraboloid z = x2 + y 2.
◦ The lower limit of integration with respect to z is the equation of the surface where
the arrow enters W . This surface is the xy -plane, where z = 0, so:
the lower limit with respect to z is: z =
◦ The upper limit of integration with respect to z is the equation of the surface where
the arrow leaves W . This surface is the paraboloid z = x2 + y 2, so:
the upper limit with respect to z is: z =
˚W
f (x, y , z) dV =
¨D
[ ˆ z=z=
f (x, y , z) dz
]dA
(D is the 2-d projection (“shadow”) of W on the xy -plane.)
BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals 2
(Q2) To find the y - and x-limits of integration, use the region D (the 2-d projection of W onto
the xy -plane) in Diagram 1B. Find the x- and y -limits using the same methods you used for
finding limits of integration for double integrals.
◦ The lower limit with respect to y is: y =
◦ The upper limit with respect to y is: y =
◦ The lower limit with respect to x is: x =
◦ The upper limit with respect to x is: x =
(Q3) Add the x-, y -, and z-limits of integration from (Q1) and (Q2) to the integral in Model 1.
Model 2: Finding Limits of Integration, Part II
W is the 3-d region in the first octant (x, y , z ≥ 0) bounded by the surfaces z = 0, x = 0,
x = 3, y = 4, and z =√y :
The projections (“shadows”) of W onto the coordinate planes are the 2-d regions:
projection of W onto the xy -plane
projection of W onto the xz-plane
projection of W onto the yz-plane
BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals 3
Critical Thinking Questions
In this section, you will find the limits of integration for the triple integral:
˚W
f (x, y , z) dV (using three different orders of integration)
where W is the 3-d region in the first octant (x, y , z ≥ 0) bounded by the surfaces z = 0, x = 0,
x = 3, y = 4, and z =√y .
(Q4) Go to Model 2: On the 3-d region W , label the five bounding surfaces with their equations.
(Q5) To find the limits of integration for the integral˝Wf (x, y , z) dV using the order of integration
dV = dz dx dy :
(a) Begin by finding the limits with respect to z . To do this, draw an arrow through the
region W parallel to the z-axis, beginning below W and ending above W . The lower
z-limit is given by the equation z = S1(x, y) of the surface where the arrow enters W .
The upper z-limit is given by the equation z = S2(x, y) of the surface where the arrow
leaves W . Using these limits:˚W
f (x, y , z) dV =
¨Dxy
(ˆ z=z=
f (x, y , z) dz
)dA
(b) Identify the 2-d region in Model 2 that represents the projection of W onto the xy -plane.
(This is the “shadow” of W that you would see on the xy -plane when looking down the
z-axis.) On this region, label the boundary curves with their equations.
(c) On this 2-d region Dxy , use the methods for finding limits of integration for double
integrals to find the remaining limits:˚W
f (x, y , z) dV =
ˆ y=y=
ˆ x=x=
ˆ z=z=
f (x, y , z) dz dx dy
(Q6) Use the process outlined in (Q5) to find the limits of integration using the order of integration
dV = dx dz dy . After finding the limits with respect to x , you will need to identify the 2-d
region in Model 2 that represents the projection of W onto the yz-plane, and use this to find
the limits of integration with respect to y and z .˚W
f (x, y , z) dV =
¨Dyz
(ˆ x=x=
f (x, y , z) dx
)dA
=
ˆ y=y=
ˆ z=z=
ˆ x=x=
f (x, y , z) dx dz dy
(Q7) Again, use the process outlined in (Q5) to find the limits of integration using the order of
integration dV = dy dx dz .
˚W
f (x, y , z) dV =
BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals 4
Model 3: Triple Integrals and Volume
W is the 3-d region in the first octant bounded
the planes:
x = 0, y = 0, z = 0, and 6x + 3y + 2z = 6
Critical Thinking Questions
In this section, you will see how triple integrals can be used to measure volume.
(Q8) Set up a double integral˜Df (x, y) dA that gives the volume of the 3-d region W , using the
order of integration dA = dy dx .
(Q9) Set up the triple integral˝WdV =
˝W
1 dV , using the order of integration dV = dz dy dx .
Do not evaluate this integral.
(Q10) Now, integrate your triple integral from (Q9) only with respect to z . Compare the result of
this to the double integral in (Q8).
(Q11) Complete this statement:
“The triple integral˝WdV =
˝W
1 dV measures the of a regionW .”
(Q12) Integration is a process of “chopping and adding”. With this in mind, use what you know
about the volume element dV to explain your answer to (Q11).
BSU Math 275 (Ultman) Worksheet 3.3: Introduction to Triple Integrals 5
Model 4: Using Limits of Integration to Sketch Regions of Inte-
gration
˚WA
dV =
ˆ 2−2
ˆ 4−4
ˆ 50
dz dy dx
Integral A
˚WB
dV =
ˆ 2−2
ˆ √4−x2−√4−x2
ˆ 50
dz dy dx
Integral B
Critical Thinking Questions
In this section, you use limits of integration to sketch regions of integration.
(Q13) Use the limits of integration from Integral A in Model 4 to sketch the 3-d region of integration
WA. What is the shape of this region?
(Q14) Use the limits of integration from Integral B in Model 4 to sketch the 3-d region of integration
WB. What is the shape of this region?
(Q15) Use the triple integrals Integral A and Integral B to compute the volumes of the regions WAand WB. Do you get what you expect?
