91Ó°ÊÓ

Converge. Evaluate the integrals without using tables. $$\int_{1}^{\infty} \frac{d x}{x^{1.001}}$$

The instructions for the integrals in Exercise have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right|.\) b. Evaluate the integral directly and find \(\left|E_{T}\right|.\) c. Use the formula \(\left(\left|E_{T}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{T}\right|\) as a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right|.\) b. Evaluate the integral directly and find \(\left|E_{S}\right|.\) c. Use the formula ( \(\left|E_{S}\right| /\) (true value)) \(\times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$\int_{-1}^{1}\left(x^{2}+1\right) d x$$

Converge. Evaluate the integrals without using tables. $$\int_{-8}^{1} \frac{d x}{x^{1 / 3}}$$

Converge. Evaluate the integrals without using tables. $$\int_{-\infty}^{2} \frac{2 d x}{x^{2}+4}$$

Estimate the minimum number of sub intervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises 11-18 are the integrals from Exercises 1-8. ) $$\int_{1}^{2} \frac{1}{s^{2}} d s$$

A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth \(h(x)\) of the water at 5 -ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with \(n=10\) applied to the integral $$V=\int_{0}^{50} 30 \cdot h(x) d x$$ $$\begin{array}{cccc} \hline \text { Position (ft) } & \text { Depth (ft) } & \text { Position (ft) } & \text { Depth (ft) } \\ x & h(x) & x & h(x) \\ \hline 0 & 6.0 & 30 & 11.5 \\ 5 & 8.2 & 35 & 11.9 \\ 10 & 9.1 & 40 & 12.3 \\ 15 & 9.9 & 45 & 12.7 \\ 20 & 10.5 & 50 & 13.0 \\ 25 & 11.0 & & \\ \hline \end{array}$$

The error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is no elementary expression for the antiderivative of \(e^{-r^{2}}\) a. Use Simpson's Rule with \(n=10\) to estimate erf (1). b. \(\operatorname{In}[0,1].\) $$\left|\frac{d^{4}}{d t^{4}}\left(e^{-t^{2}}\right)\right| \leq 12.$$ Give an upper bound for the magnitude of the error of the estimate in part (a).

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{x^{2}+x}{x^{4}-3 x^{2}-4} d x$$

In Exercises \(33-38,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$\int \frac{16 x^{3}}{4 x^{2}-4 x+1} d x$$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{2} \frac{d x}{1-x^{2}}$$