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Various exercises relating to numerical integration.
Consider the definite integral .
The trapeziod rule with gives the approximation
Simpson’s rule with gives the approximation
The exact value of the integral is
When estimating the integral below using Simpson’s rule, what is the minimum
number of intervals that would be required to guarantee that the approximation error
does not exceed ? (Enter the smallest value which you know is correct.)
Find such that the error in approximating the given definite integral is less than
The trapezoid rule: (Enter your answer as the exact result of your
calculation; do not round or approximate.)
Simpson’s rule: (Enter your answer as the exact result of your calculation;
do not round or approximate.)
How many equally spaced intervals are sufficient for the trapezoidal rule to estimate
the value of the following integral with an error less than or equal to ? (Enter the
smallest value which you know is correct.)
The second derivative of the integrand is
. Since , it follows that
By the trapezoid rule error formula, the error satisfies
where is the number of intervals.