Double Integrals for Rectangles:
The center point method:
Presume that we need to find the integral of a function, f(x, y) on a rectangle:
R = {(x, y) :a ≤ x ≤ b, c ≤ y ≤ d}.
In calculus you erudite to do this by an iterated integral:
You also must have learned that the integral is the limit of the Riemann sums of the function as the size of the sub-rectangles goes to zero. This signifies that the Riemann sums are approximations of the integral which is exactly what we need for numerical methods.
For a rectangle R we start by subdividing into smaller sub-rectangles {Rij} in a systematic way. We will divide [a, b] into m subintervals as well as [c, d] into n subintervals. Afterwards Rij will be the “intersection” of the i-th subinterval in [a, b] with the j-th subinterval of [c, d]. In this way the complete rectangle is subdivided into mn sub-rectangles numbered as in given figure.
A Riemann sum utilizing this subdivision would have the form:
Subdivision of the rectangle R = [a, b] × [c, d] into sub-rectangles Rij
Where Aij= ΔxiΔyj is the area of Rij, and x*ij is a point in Rij. The theory of integrals notify us that if f is continuous then this calculation will converge to the same number no matter how we choose x*ij. For illustration we could choose x*ij to be the point in the lower left corner of Rij and the sum would still converge as the size of the sub-rectangles goes to zero. Nevertheless in practice we wish to choose x*ij in such a way to make S as accurate as possible even when the sub-rectangles are not very small. The noticeable choice for the best point in Rij would be the center point. The center point is most probable of all points to have a value of f close to the average value of f. If we denote the center points by cij then the sum becomes:
Note that if the subdivision is consistently spaced then Δx ≡ (b − a)/m and Δy ≡ (d − c)/n and therefore in that case:
The four corners method:
An extra good idea would be to take the value of f not only at one point however as the average of the values at several points. An clear choice would be to evaluate f at all four corners of each Rijthen average those. If we note that the lower left corner is (xi, yj), the upper left is (xi, yj+1) the lower right is (xi+1, yi) as well as the upper right is (xi+1, yi+1) then the corresponding sum will be:
Which we will call the four-corners method. If the sub-rectangles are consistently spaced then we can simplify this expression. Notice that f(xi, yj) gets counted multiple times depending on where (xi, yj) is located. For illustration if (xi, yj) is in the interior of R then it is the corner of 4 sub-rectangles.
Consequently the sum becomes:
Where A = ΔxΔy is the area of the sub-rectangles. We are able to think of this as a weighted average of the values of f at the grid points (xi, yj). The weights utilized are represented in the matrix:
We could execute the four-corner method by forming a matrix (fij) of f values at the grid points then doing entry-wise multiplication of the matrix with the weight matrix. Then the integral would be acquired by summing all the entries of the resulting matrix and multiplying that by A/4. The formula would be:
Notice that the four-corner method coincides among applying the trapezoid rule in each direction.
Therefore it is in fact a double trapezoid rule.
The double Simpson method:
The subsequent improvement one might make would be to take an average of the center point sum Cmnand the four corners sum Fmn. Nevertheless a more standard way to acquire a more accurate method is the Simpson double integral. It is acquire by applying Simpson’s rule for single integrals to the iterated double integral. The resulting method needs that both m and n be even numbers and the grid are evenly spaced. If this is the case we summarize the values f(xi, yj) with weights represented in the matrix:
The amount of the weighted values is multiplied by A/9 and the formula is:
Mat lab has a built in command for double integrals on rectangles dblquad(f,a,b,c,d). Here is an instance
> f = inline(’sin(x.*y)./sqrt(x+y)’,’x’,’y’)> I = dblquad(f,0.5,1,0.5,2)
Below is a Mat lab function which will produce the matrix of weights needed for Simpson’s rule for double integrals. It utilize the function mysimpweights.
function W = mydblsimpweights(m,n)% Produces the m by n matrix of weights for Simpson’s rule% for double integrals% Inputs: m -- number of intervals in the row direction.% must be even.% n -- number of intervals in the column direction.% must be even.% Output: W -- a (m+1)x(n+1) matrix of the weightsif rem(m,2)~=0 | rem(n,2)~=0error(’m and n must be even’)endu = mysimpweights(m);v = mysimpweights(n);W = u*v’
Latest technology based Matlab Programming Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Matlab Programming help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Matlab Programming, project ideas and tutorials. We provide email based Matlab Programming help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Matlab Programming. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Matlab Programming Homework help and assignment help services. They use their experience, as they have solved thousands of the Matlab Programming assignments, which may help you to solve your complex issues of Matlab Programming. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
For filling in transformers good oil should withstand at least 40 kV (in the above condition (vi- a)) for one minute.
tutorsglobe.com hydrolytic fermentative stage assignment help-homework help by online recycling of waste tutors
tutorsglobe.com commonly available medicinal plants assignment help-homework help by online medicinal plants including microbes tutors
There is an assumption in accounting that money, that is the unit of measurement, will stay stable over time. However, this is an unrealistic assumption as the value of money changes. Generally, it is inflation that is the culprit.
tutorsglobe.com amoebiasis assignment help-homework help by online protozoan microbiology tutors
tutorsglobe.com monetary policy of money assignment help-homework help by online monetary policy tutors
tutorsglobe.com autogamy assignment help-homework help by online self pollination tutors
Mutation tutorial all along with the key concepts of Causes of mutation, Spontaneous mutation, Induced mutation, Classifying Mutations, Structural Effects, Functional Effects, Harmful mutations and Beneficial Mutations
Principles of Infrared Spectroscopy tutorial all along with the key concepts of Principles of Infrared Spectroscopy, Intensity and Energy Level of Absorption in IR-Spectroscopy , Instrumentation of Infrared Spectroscopy, Infrared Spectrum and Interpretation of Infrared Spectra
Avail impeccable Basic Definitions of Statistics Assignment Help and our PhD experts will solve it keeping 100% precision in easy to understand manner.
tutorsglobe.com uses of coordination compounds assignment help-homework help by online coordination compounds tutors
Rubber Elasticity tutorial all along with the key concepts of Natural rubber, Elasticity of Rubber, Thermodynamics of Rubber Elasticity, Main uses of Natural Rubber
tutorsglobe.com recent theories of wages assignment help-homework help by online wages tutors
tutorsglobe.com shift in demand assignment help-homework help by online shift in demand and supply tutors
tutorsglobe.com diffusion pressure deficit assignment help-homework help by online plasmolysis tutors
1959088
Questions Asked
3689
Tutors
1455058
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!