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## Double Integrals for Rectangles

Double Integrals for Rectangles::The center point methodPresume 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 {R

_{ij}} in a systematic way. We will divide [a, b] into m subintervals as well as [c, d] into n subintervals. Afterwards R_{ij}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 R

_{ij}Where A

_{ij}= Δx_{i}Δy_{j}is the area of R_{ij}, and x*_{ij}is a point in R_{ij}. 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 R_{ij}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 R_{ij}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 c_{ij}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 methodAn 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 (x

_{i}, y_{j}), the upper left is (x_{i}, y_{j+1}) the lower right is (x_{i+1}, y_{i}) as well as the upper right is (x_{i+1}, y_{i+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 (x

_{i}, y_{j}) is located. For illustration if (x_{i}, y_{j}) 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 (x

_{i}, y_{j}). The weights utilized are represented in the matrix:We could execute the four-corner method by forming a matrix (f

_{ij}) 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 methodThe 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(x

_{i}, y_{j}) 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 weights

if rem(m,2)~=0 | rem(n,2)~=0

error(’m and n must be even’)

end

u = mysimpweights(m);

v = mysimpweights(n);

W = u*v’

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