91Ó°ÊÓ

The instructions for the integrals in Exercises \(1-10\) 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 (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. 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. $$ \int_{1}^{2} x d x $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{x^{2}}{x^{2}+1} d x $$

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{2 x+2}{x^{2}-2 x+1}$$

Lifetime of light bulbs A manufacturer of light bulbs finds that the mean lifetime of a bulb is 1200 hours. Assume the life of a bulb is exponentially distributed. $$ \begin{array}{l}{\text { a. Find the probability that a bulb will last less than its guaranteed }} \\ {\text { lifetime of } 1000 \text { hours. }} \\\ {\text { b. In a batch of light bulbs, what is the expected time until half }} \\\ {\text { the light bulbs in the batch fail? }}\end{array} $$

Failure time The time between failures of a photocopier is exponentially distributed. Half of the copiers at a university require service during the first 2 years of operations. If the university purchased 150 copiers, how many do you expect to require service during the first year of their operation?

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. $$\int_{2}^{\infty} \frac{d x}{\sqrt{x^{2}-1}}$$