Exact Simpson's Rule a. Use Simpson's Rule to approximate ∫0^4 x^3 d x using two subintervals (n

step 1 All right, problem is 75. The function integrating from 0 to 4 of x cubed dx. So part A using tw

Exact Simpson's Rule a. Use Simpson's Rule to approximate...

Exact Simpson's Rule a. Use Simpson's Rule to approximate $\int_{0}^{4} x^{3} d x$ using two subintervals $(n=2) ;$ compare the approximation to the value of the integral. b. Use Simpson's Rule to approximate $\int_{0}^{4} x^{3} d x$ using four subintervals $(n=4) ;$ compare the approximation to the value of the integral. c. Use the error bound associated with Simpson's Rule given in Theorem 8.1 to explain why the approximations in parts (a) and (b) give the exact value of the integral. d. Use Theorem 8.1 to explain why a Simpson's Rule approximation using any (even) number of subintervals gives the exact value of $\int_{a}^{b} f(x) d x,$ where $f(x)$ is a polynomial of degree 3 or less.

Exact Simpson's Rule
a. Use Simpson's Rule to approximate $\int_{0}^{4} x^{3} d x$ using two subintervals $(n=2) ;$ compare the approximation to the value of the integral.
b. Use Simpson's Rule to approximate $\int_{0}^{4} x^{3} d x$ using four subintervals $(n=4) ;$ compare the approximation to the value of the integral.
c. Use the error bound associated with Simpson's Rule given in Theorem 8.1 to explain why the approximations in parts (a) and (b) give the exact value of the integral.
d. Use Theorem 8.1 to explain why a Simpson's Rule approximation using any (even) number of subintervals gives the exact value of $\int_{a}^{b} f(x) d x,$ where $f(x)$ is a polynomial of degree 3 or less.

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Key Concepts

Exactness for Cubic Polynomials

Simpson's Rule is exact for polynomials of degree three or less due to the way its quadratic interpolation inherently captures the behavior of cubic functions exactly over any interval. Since the rule utilizes parabolic approximations, when applied to cubic or lower degree polynomials, the approximation error vanishes, resulting in the exact value of the definite integral.

Error Bound for Simpson's Rule

The error bound associated with Simpson's Rule provides an upper limit for the error in the approximation by relating it to the maximum value of the fourth derivative of the integrand over the interval. This bound gives theoretical guarantees on the accuracy of the approximation, depending on the number of subintervals chosen and the behavior of the function's fourth derivative.

Simpson's Rule

Simpson's Rule is a numerical integration technique that approximates the value of a definite integral by using a quadratic interpolant over subintervals of the domain. This method divides the integration interval into an even number of subintervals and fits second-degree polynomials to the subintervals, combining these approximations to yield an overall estimate of the integral.

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