The rate, r, at which people get sick during an epidemic of the flu can be approximated

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 4 0 6xdx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 3 1 5 dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 3 0 t3 dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 2 0 (12x2 + 1) dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 2 0 (3t2 + 4t + 3) dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 2 1 1 x dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 4 1 1 x dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 1 0 (6q2 + 4) dq

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 5 0 3x2 dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 1 0 2ex dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 2 1 5t3 dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 3 1 6x2 dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 1 0 (y2 + y4) dy

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 2 1 1 x2 dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 5 2 (x3 x2) dx

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 1 0 e0.2t dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 1 1 cos t dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < /4 0 (sin t + cos t) dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 3 0 e0.05t dt

Using the Fundamental Theorem, evaluate the definite integrals in Problems 120 exactly. < 9 4 xdx

Find the exact area of the region bounded by the x-axis and the graph of y = x3 x.

Use the Fundamental Theorem of Calculus to find the average value of f(x) = e0.5x between x = 0 and x = 3. Show the average value on a graph of f(x).

Use the Fundamental Theorem to determine the value of b if the area under the graph of f(x) = x2 between x = 0 and x = b is equal to 100. Assume b > 0.

Use the Fundamental Theorem to determine the value of b if the area under the graph of f(x) = 4x between x = 1 and x = b is equal to 240. Assume b > 1.

If t is in years, and t = 0 is January 1, 2005, worldwide energy consumption, r, in quadrillion (1015) BTUs per year, is modeled by r = 462e0.019t . (a) Write a definite integral for the total energy use between the start of 2005 and the start of 2010. (b) Use the Fundamental Theorem of Calculus to evaluate the integral. Give units with your answer.

Oil is leaking out of a ruptured tanker at the rate of r(t) = 50e0.02t thousand liters per minute. (a) At what rate, in liters per minute, is oil leaking out at t = 0? At t = 60? (b) How many liters leak out during the first hour?

(a) Between 2000 and 2010, ACME Widgets sold widgets at a continuous rate of R = R0e0.125t widgets per year, where t is time in years since January 1, 2000. Suppose they were selling widgets at a rate of 1000 per year on January 1, 2000. How many widgets did they sell between 2000 and 2010? How many did they sell if the rate on January 1, 2000 was 1,000,000 widgets per year? (b) In the first case (1000 widgets per year on January 1, 2000), how long did it take for half the widgets in the ten-year period to be sold? In the second case (1,000,000 widgets per year on January 1, 2000), when had half the widgets in the ten-year period been sold? (c) In 2010, ACME advertised that half the widgets it had sold in the previous ten years were still in use. Based on your answer to part (b), how long must a widget last in order to justify this claim?

Decide if the improper integral = 0 e2t dt converges, and if so, to what value, by the following method. (a) Use a computer or calculator to find =b 0 e2t dt for b = 3, 5, 7, 10. What do you observe?Make a guess about the convergence of the improper integral. (b) Find =b 0 e2t dt using the Fundamental Theorem. Your answer will contain b. (c) Take a limit as b . Does your answer confirm your guess?

In this problem, you will show that the following improper integral converges to 1: < 1 1 x2 dx. (a) Use the Fundamental Theorem to find =b 1 1/x2 dx. Your answer will contain b. (b) Now take the limit as b . What does this tell you about the improper integral?

(a) Graph f(x) = ex2 and shade the area represented by the improper integral = ex2 dx. (b) Use a calculator or computer to find =a a ex2 dx for a = 1, a = 2, a = 3, a = 5. (c) The improper integral = ex2 dx converges to a finite value. Use your answers from part (b) to estimate that value.

Graph y = 1/x2 and y = 1/x3 on the same axes.Which do you think is larger: = 1 1/x2 dx or = 1 1/x3 dx? Why?

At a time t hours after taking a tablet, the rate at which a drug is being eliminated is r(t) = 50&e0.1t e0.2t' mg/hr. Assuming that all the drug is eventually eliminated, calculate the original dose.

The rate, r, at which people get sick during an epidemic of the flu can be approximated by r = 1000te0.5t , where r is measured in people/day and t is measured in days since the start of the epidemic. (a) Write an improper integral representing the total number of people that get sick. (b) Use a graph of r to represent the improper integral from part (a) as an area.

An island has a carrying capacity of 1 million rabbits. (That is, no more than 1 million rabbits can be supported by the island.) The rabbit population is two at time t = 1 day and grows at a rate of r(t) thousand rabbits/day until the carrying capacity is reached. For each of the following formulas for r(t), is the carrying capacity ever reached? Explain your answer. (a) r(t) = 1/t2 (b) r(t) = t (c) r(t) = 1/t