Math 251 Final Review Pack

1 double integrals over rectangles recall area and

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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 x-axis are constant and whose boundaries along the y-axis are functions of x (variable) Type 2 Region: A region whose boundaries along the y-axis are constant and whose boundaries along the x-axis 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 xy-plane and the x-axis φ is the angle subtended by the line ρ and the z-axis Because: 43 In spherical coordinates: Half-plane Half-cone above the xy-plane Half-cone below the xy-plane 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.

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