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Lecture 4: Multiple Integrals
• Recall physical interpretation of a 1D integral as area under curve
• Divide domain (a,b) into n strips, each of width δrk , k=1,2...,n
• As we get
• With value of function constant within each strip
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2D Integrals
• Definition
• Cartesian Grid
or alternatively (equivalently)
c
A
a b
y
x
• Divide into n sub-regions, eachof area δAk ,k=1,2...,n
• For small δAk, value of function is constant across the patch
•
• Complicated regions may needcarving up
d
C
B D
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Physical Interpretation
• 2D integrals can beinterpreted as a volume
• The rectangular regionhere represents anexample domain ofintegration
• The height of surface isdefined by the value of
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Example• R is a region bounded by y=x2, x=2, y=1
• Calculate
• Order of integration unimportant for well-behaved functions
1 2
4
1
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Another Example
x
y
(0,1)
(1,0)
Exercise for student: obtain same answer by integrating in other order: e.g., y first.
Do x first
Then do y
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3D Integrals• Divide volume V into sub-volumes each of volume
• Definition as per 2D but generalised to add up n volumes (rather than n areas)
• Using Cartesian grid
• Note care is needed if V has dimples
a slice of volume
and its projectiononto the xy-plane
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Simple Example (Cartesian)
• Find the volume V bounded by the parabolic cylinder
and the planes
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Cylindrical Polar Coordinates
Note r is perpendicular distance from the cylinder axis; φ is the azimuthal angle
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Simple Example (Cylindrical)• Find the volume V of a cylinder radius r, height h
• In Cartesians:
• In Cylindrical Polars
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Harder Example (Cylindrical)• Find the moment of inertiaabout the z axis of the solid thatlies below the paraboloid
inside the cylinder
and above the xy plane, havinga density function
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Spherical Polar Coordinates
Azimuthal Angle
“longitude eastwards”
Co-latitude
Note r is now a radial
distance from origin
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Simple Example (Spherical)• Find the volume V of a sphere radius r
• In Cartesians:
•
• In Spherical Polars
Projection ontoz=0 planeHeight of this strip is
x
y
Exercise forstudent!
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Harder Example (Spherical)• Find the volume V that lies insidethe sphere
and outside the cone
• In spherical polars