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82 LECTURE 22. INTEGRATION: MIDPOINT AND SIMPSON’S RULES
In Matlab there is a built-in command for definite integrals: quad(f,a,b) where the f is an inline
function and a and b are the endpoints. Here quad stands for quadrature, which is a term for
numerical integration. The command uses “adaptive Simpson quadrature”, a form of Simpson’s
rule that checks its own accuracy and adjusts the grid size where needed. Here is an example of its
usage:
> f = inline(’x.^(1/3).*sin(x.^3)’)
> I = quad(f,0,1)
Exercises
22.1 Using formulas (22.1) and (22.2), for the integral
∫ 2
1
√
x dx calculate M4 and S4 (by hand, but
use a calculator). Find the percentage error of these approximations, using the exact value.
Compare with exercise 21.1.
22.2 Write a function program mymidpoint that calculates the midpoint rule approximation for
∫
f on the interval [a, b] with n subintervals. The inputs should be f , a, b and n. Use your
program on the integral
∫ 2
1
√
x dx to obtain M4 and M100. Compare these with the previous
exercise and the true value of the integral.
22.3 Write a function program mysimpson that calculates the Simpson’s rule approximation for
∫
f on the interval [a, b] with n subintervals. It should call the program mysimpweights to
produce the coefficients. Use your program on the integral
∫ 2
1
√
x dx to obtain S4 and S100.
Compare these with the previous exercise and the true value.
82 LECTURE 22. INTEGRATION: MIDPOINT AND SIMPSON’S RULES
In Matlab there is a built-in command for definite integrals: quad(f,a,b) where the f is an inline
function and a and b are the endpoints. Here quad stands for quadrature, which is a term for
numerical integration. The command uses “adaptive Simpson quadrature”, a form of Simpson’s
rule that checks its own accuracy and adjusts the grid size where needed. Here is an example of its
usage:
> f = inline(’x.^(1/3).*sin(x.^3)’)
> I = quad(f,0,1)
Exercises
22.1 Using formulas (22.1) and (22.2), for the integral
∫ 2
1
√
x dx calculate M4 and S4 (by hand, but
use a calculator). Find the percentage error of these approximations, using the exact value.
Compare with exercise 21.1.
22.2 Write a function program mymidpoint that calculates the midpoint rule approximation for
∫
f on the interval [a, b] with n subintervals. The inputs should be f , a, b and n. Use your
program on the integral
∫ 2
1
√
x dx to obtain M4 and M100. Compare these with the previous
exercise and the true value of the integral.
22.3 Write a function program mysimpson that calculates the Simpson’s rule approximation for
∫
f on the interval [a, b] with n subintervals. It should call the program mysimpweights to
produce the coefficients. Use your program on the integral
∫ 2
1
√
x dx to obtain S4 and S100.
Compare these with the previous exercise and the true value.
