Chapter 5

Compute the definite integral and interpret the result in terms of areas. $$ \int_{1}^{4}(x-3 \ln x) d x $$

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{2} 2^{x} d x $$

Compute the definite integral \(\int_{0}^{4} \cos \sqrt{x} d x\) and interpret the result in terms of areas.

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{2}(1.03)^{t} d t $$

Find the area between the graph of \(y=x^{2}-2\) and the \(x\) -axis, between \(x=0\) and \(x=3\).

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{3} \ln x d x $$

Use an integral to find the specified area. Under \(y=6 x^{3}-2\) for \(5 \leq x \leq 10\).

Use a calculator or computer to evaluate the integral. $$ \int_{1.1}^{1.7} e^{t} \ln t d t $$

Use an integral to find the specified area. Under \(y=2 \cos (t / 10)\) for \(1 \leq t \leq 2\)

The amount of waste a company produces, \(W\), in tons per week, is approximated by \(W=3.75 e^{-0.008 t}\), where \(t\) is in weeks since January 1,2005 . Waste removal for the company costs \(\$ 15 /\) ton. How much does the company pay for waste removal during the year \(2005 ?\)

Use a calculator or computer to evaluate the integral. $$ \int_{-3}^{3} e^{-t^{2}} d t $$

Use an integral to find the specified area. Under \(y=5 \ln (2 x)\) and above \(y=3\) for \(3 \leq x \leq 5\).

Use an integral to find the specified area. Between \(y=\sin x+2\) and \(y=0.5\) for \(6 \leq x \leq 10\).

Use an integral to find the specified area. Between \(y=\cos x+7\) and \(y=\ln (x-3), 5 \leq x \leq 7\)

Use an integral to find the specified area. Above the curve \(y=x^{4}-8\) and below the \(x\) -axis.

Use an integral to find the specified area. Above the curve \(y=-e^{x}+e^{2(x-1)}\) and below the \(x\) axis, for \(x \geq 0\).

Use an integral to find the specified area. Between \(y=\cos t\) and \(y=\sin t\) for \(0 \leq t \leq \pi\)

At the site of a spill of radioactive iodine, radiation levels were four times the maximum acceptable limit, so an evacuation was ordered. If \(R_{0}\) is the initial radiation level (at \(t=0\) ) and \(t\) is the time in hours, the radiation level \(R(t)\), in millirems/hour, is given by $$ R(t)=R_{0}(0.996)^{t} $$ (a) How long does it take for the site to reach the acceptable level of radiation of \(0.6\) millirems/hour? (b) How much total radiation (in millirems) has been emitted by that time?

If you jump out of an airplane and your parachute fails to open, your downward velocity (in meters per second) \(t\) seconds after the jump is approximated by $$ v(t)=49\left(1-(0.8187)^{t}\right) $$ (a) Write an expression for the distance you fall in \(T\) seconds. (b) If you jump from 5000 meters above the ground, estimate, using trial and error, how many seconds you fall before hitting the ground.