Write a program to calculate double integral numerically.
Input: Given the following integral. where Output: 3.915905
Explanation and Approach:
- We need to decide what method we are going to use to solve the integral.
In this example, we are going to use Simpson 1/3 method for both x and y integration.
To do so, first, we need to decide the step size. Let h be the step size for integration with respect to x and k be the step size for integration with respect to y.
We are taking h=0.1 and k=0.15 in this example.
Refer for Simpson 1/3 rule
- We need to create a table which consists of the value of function f(x, y) for all possible combination of all x and y points.
x\y y0 y1 y2 …. ym x0 f(x0, y0) f(x0, y1) f(x0, y2) …. f(x0, ym) x1 f(x1, y0) f(x1, y1) f(x1, y2) …. f(x1, ym) x2 f(x2, y0) f(x2, y1) f(x2, y2) …. f(x2, ym) x3 f(x3, y0) f(x3, y1) f(x3, y2) …. f(x3, ym) …. …. …. …. …. …. …. …. …. …. …. …. xn f(xn, y0) f(xn, y1) f(xn, y2) …. f(xn, ym)
In the given problem,
x0=2.3 x2=2.4 x3=3.5 y0=3.7 y1=3.85 y2=4 y3=4.15 y4=4.3
- After generating the table, we apply Simpson 1/3 rule (or whatever rule is asked in the problem) on each row of the table to find integral wrt y at each x and store the values in an array ax.
- We again apply Simpson 1/3 rule(or whatever rule asked) on the values of array ax to calculate the integral wrt x.
Below is the implementation of the above code:
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