91Ó°ÊÓ

Approximate the integral to three decimal places using the indicated rule. $$ \int_{0.1}^{0.5} \frac{\cos x}{x} d x \text { ; Simpson's rule; } n=4 $$

Evaluate \(\int_{0}^{1} \frac{d x}{1+x^{2}}\) exactly and show that the result is \(\pi / 4\). Then, find the approximate value of the integral using the trapezoidal rule with \(n=4\) subdivisions. Use the result to approximate the value of \(\pi\).

Approximate \(\int_{2}^{4} \frac{1}{\ln x} d x\) using the trapezoidal rule with eight subdivisions to four decimal places.

Use the trapezoidal rule with four subdivisions to estimate \(\int_{0}^{0.8} x^{3} d x\) to four decimal places.

Use the trapezoidal rule with four subdivisions to estimate \(\int_{0}^{0.8} x^{3} d x\). Compare this value with the exact value and find the error estimate.

Using Simpson's rule with four subdivisions, find \(\int_{0}^{\pi / 2} \cos (x) d x\)

Find an upper bound for the error in estimating \(\int_{4}^{5} \frac{1}{(x-1)^{2}} d x\) using the trapezoidal rule with seven subdivisions.

Find an upper bound for the error in estimating \(\int_{0}^{3}\left(6 x^{2}-1\right) d x\) using Simpson's rule with \(n=10\) steps.

Find an upper bound for the emor in estimating \(\int_{2}^{5} \frac{1}{x-1} d x\) using Simpson's nule with \(n=10\) steps.

Estimate the minimum number of subintervals needed to approximate the integral \(\int_{2}^{3}\left(2 x^{3}+4 x\right) d x\) with an error of magnitude less than 0.0001 using the trapezoidal rule.