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Repeat Exercise 23 for the integral $ \displaystyle \int_{-1}^1 \sqrt{4 - x^3}\ dx $.

a. $\left|f^{\prime \prime}(x)\right| \leq 5$

b. Midpoint rule approximation is 3.9958041521928

c. Error bound is 0.0066

d. 3.99549 using Simpson's Rule with $n=50$

e. Actual error is less than that found in part (c)

f. $f^{\prime \prime \prime \prime}(x)$ is bounded between $-0.2$ and $-0.4$ in $(-1,1)$

g. Simpsons rule approximation with $n=10$ is 3.9954497901518

h. Error bound is 0.000007111111

i. Actual error in approximation is more than the theoretical error calculated

j. n should be 712 or more for the simpsons approximation to have an error less than 0.0001

Integration Techniques

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University of Wisconsin - Madison

Integration Techniques