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Unformatted text preview: ed by Provided that this limit exists
We say that f is integrable if this limit exists
If , then is the volume under the surface above R
Definition (Midpoint Rule): We define the midpoint of a rectangle as the point
then the midpoint rule is defined as 33 and Where this double Riemann sum correspond to this choice of sample point
Recall:
In single variable calculus: Now for multivariable calculus: Properties:
1)
2)
3) If , 34 15.2 Iterated Integrals
Rectangles:
Theorem (Fubini’s Theorem): If f is continuous on R, then The iterated integral is the integral in the brackets and can be treated as a function of x.
Steps of Calculating
1) Compute the function of x,
2) Compute 35 15.3 Double Integrals Over General Regions Type 1 Region: A region whose boundaries along the xaxis are constant and whose boundaries
along the yaxis are functions of x (variable)
Type 2 Region: A region whose boundaries along the yaxis are constant and whose boundaries
along the xaxis are functions of y (variable)
Provided the functions are continuous along the domain:
For Type 1 Regions: For Type 2 Regions: 36 Changing the Order of Integration
Consider the following double integral: Using change of order: We can now rewrite our integral: As we can see this is much easier to evaluate
In general, we need to graph the bounds of the variable which is making the integral difficult to
solve and analyze it to find different bounds of integration to make the integral easier to solve. 37 15.4 Double Integrals in Polar Regions
Recall our change of variables in polar coordinates Recall the area of a slice We divide the interval
subintervals into m subintervals and we also divide the interval The Riemann sum is: We simply use the change of variables for polar coordinates 38 into n We can write the integral as We can also see that there are both type 1 and type 2 regions in this set of coordinates 39 15.5 Applications of Double Integrals
Mass
 Constant density mass = density + area
Variable density density function Electric Charge
Charge function Center of Mass
SEE TEXTBOOK 40 15.7 Triple Integrals  Divide into sub rectangular boxes
Take sample points
Send the number of boxes to infinity Computing triple integrals is very similar to computing iterated integrals, except do the process
twice
Theorem (Fubini’s Theorem): As with double integrals, Fubini’s Theorem also holds for triple
integrals provided that the function is continuous over the solid
Particular Cases: Applications: Mass
Volume
Moments of Inertia
Center of Mass 41 15.8 Triple Integrals in Cylindrical Coordinates
Recall from polar coordinates For all nonzero values for r
Cylindrical coordinates: Triple Integrals in Cylindrical Coordinates Essentially, covert x,y to polar coordinates 42 15.9 Triple Integrals in Spherical Coordinates
ρ is the radius from the origin to some point
Θ is the angle subtended by the projection of the line ρ on the xyplane and the xaxis
φ is the angle subtended by the line ρ and the zaxis Because: 43 In spherical coordinates:
Halfplane
Halfcone above the xyplane
Halfcone below the xyplane
Conside...
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This note was uploaded on 02/03/2014 for the course CMPT 150 taught by Professor Dr.anthonydixon during the Spring '08 term at Simon Fraser.
 Spring '08
 Dr.AnthonyDixon

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