Question

The following values of function f is given. Our goal is to approximate ∫¹₂ f(x)dx.

a. Compute an approximation by the composite Midpoint Rule for n=2,
b. Compute an approximation by the composite Trapezoid Rule for n=4,
c. Compute an approximation by the composite Simpsons Rule for n=2.

Answer 1

The student's question pertains to approximating the integral of a function using numerical methods like composite Midpoint, Trapezoid, and Simpson's Rules. These methods use function evaluations at specified points to estimate the area under the curve within a certain interval.

Step-by-step explanation:

The student's question involves applying numerical integration methods to approximate the integral of a function. The specific methods mentioned are the composite Midpoint Rule, composite Trapezoid Rule, and composite Simpson's Rule.

To approximate ∫¹² f(x)dx using the composite Midpoint Rule for n=2, we would divide the interval [1,2] into two equal subintervals, find the midpoint of each, evaluate the function at these midpoints, and then multiply by the width of the subintervals to find the sum of the areas of the rectangles.

For the composite Trapezoid Rule approximation with n=4, we would divide the interval [1,2] into four equal subintervals, evaluate the function at each endpoint of these intervals, and then use these values to calculate the areas of the trapezoids summing up to approximate the integral.

The composite Simpson's Rule with n=2 uses parabolic arcs instead of straight lines to approximate the graph of the function. This method requires that n be even, and it uses the function values at the endpoints and the midpoints of the subintervals in the calculation.