Consider the definite integral $\displaystyle \int _{-1}^1 x^2\ dx$.

• The trapeziod rule with $n=4$ gives the approximation $$\int _{-1}^1 x^2 \ dx \approx \answer {\frac {3}{4}}.$$
• Simpson’s rule with $n=4$ gives the approximation $$\int _{-1}^1 x^2 \ dx \approx \answer {\frac {2}{3}}.$$
• The exact value of the integral is $$\int _{-1}^1 x^2 \ dx = \answer {\frac {2}{3}}.$$

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 $2 \times 10^{-5}$? (Enter the smallest value which you know is correct.) $$\int _{-1}^0 e^{x \sqrt {6}} dx$$ $$n \geq \answer {10}.$$

Find $n$ such that the error in approximating the given definite integral is less than $0.0001$ when using:

• The trapezoid rule: $n \geq \answer {\sqrt {\frac {10000 \pi ^3}{12}}} \approx 161.$ (Enter your answer as the exact result of your calculation; do not round or approximate.)
• Simpson’s rule: $n \geq \answer {\left (\frac {10000 \pi ^5}{180}\right )^{1/4}} \approx 12.$ (Enter your answer as the exact result of your calculation; do not round or approximate.)

How many equally spaced intervals $N$ are sufficient for the trapezoidal rule to estimate the value of the following integral with an error less than or equal to $10^{-6}$? (Enter the smallest value which you know is correct.) $$\int _{-1}^1 e^{x^2-1} dx.$$ $$N \geq \answer {2000}.$$