91Ó°ÊÓ

Your engineering firm is bidding for the contract to construct the tunnel shown here. The tunnel is 300 \(\mathrm{ft}\) long and 50 \(\mathrm{ft}\) wide at the base. The cross-section is shaped like one arch of the curve \(y=25 \cos (\pi x / 50) .\) Upon completion, the tunnel's inside surface (excluding the roadway) will be treated with a waterproof sealer that costs \(\$ 2.35\) per square foot to apply. How much will it cost to apply the sealer? (Hint: Use numerical integration to find the length of the cosine curve.)

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. $$ \int \frac{x^{4}}{x^{2}-1} d x $$

In Exercises \(27-40\) , use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$ \int \frac{d t}{\tan t \sqrt{4-\sin ^{2} t}} $$

Evaluate the integrals. \(\int \sec ^{3} x \tan x d x\)

Customer service time The mean waiting time to get served after walking into a bakery is 30 seconds. Assume that an exponential density function describes the waiting times. a. What is the probability a customer waits 15 seconds or less? b. What is the probability a customer waits longer than one minute? c. What is the probability a customer waits exactly 5 minutes? d. If 200 customers come to the bakery in a day, how many are likely to be served within three minutes?

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. $$ \int \frac{9 x^{3}-3 x+1}{x^{3}-x^{2}} d x $$

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int \frac{7 d x}{(x-1) \sqrt{x^{2}-2 x-48}} $$

Evaluate the integrals. Some integrals do not require integration by parts. $$ \int \frac{\ln x}{x^{2}} d x $$

In Exercises \(35-68\) , 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. $$\int_{1 / 2}^{2} \frac{d x}{x \ln x}$$

In Exercises \(35-48\) , use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$ \int_{0}^{\ln 4} \frac{e^{t} d t}{\sqrt{e^{2 t}+9}} $$