Chapter 24

Find the average value of \(3 \sin x+5\) on the interval \([0,2 \pi]\). Do this in two ways, first geometrically and then using the Fundamental Theorem of Calculus.

(a) Using a computer or programmable calculator, find upper and lower bounds for the area under one arc of \(\cos x\) using Riemann sums. Explain how you can be sure your lower bound is indeed a lower bound and your upper bound is an upper bound. (Do not use the Fundamental Theorem of Calculus to do so.) Your upper and lower bounds should differ by no more than \(0.01\). (b) Use the Fundamental Theorem of Calculus to show that the area under one arc of the cosine curve is exactly 2 .

(a) Suppose \(f\) is an odd function. Can you determine the average value of \(f\) on \([-a, a] ?\) If so, what is the average value? (b) Suppose \(f\) is an even function. Are the following equal? If not, can you determine which is largest? Explain your answer. i. the average value of \(f\) on \([-a, a]\) ii. the average value of \(f\) on \([0, a]\) iii. the average value of \(f\) on \([-a, 0]\)

An object's velocity at time \(t, t\) in seconds, is given by \(v(t)=10 t+3\) meters per second. Find the net distance traveled from time \(t=1\) to \(t=9\). Do this in two ways. First, look at the appropriate signed area and solve geometrically, without the Fundamental Theorem. Then calculate the definite integral $$\int_{1}^{9}(10 t+3) d t$$ using the Fundamental Theorem of Calculus.

Find the average value of \(\sin x\) on \([0, \pi]\).

Use the Fundamental Theorem of Calculus to calculate \(\int_{1}^{2} t^{3} d t\).

The velocity of an object on \([0,6]\) is given by \(v(t)=-t^{2}+4 t\). (a) Find the average velocity on \([0,6]\). (b) Find the average speed on \([0,6]\).

Evaluate \(\int_{1}^{3} \frac{1}{t} d t\)

Find the area under the graph of \(y=e^{x}\) between \(x=0\) and \(x=1\).

The amount of a certain chemical in a mixture varies with time. If \(g(t)=5 e^{-t}\) is the number of grams of the chemical at time \(t\), what is the average number of grams of the chemical in the mixture on the time interval \([0,1] ?\)

Find the area under the graph of \(y=e^{-x}\) between (a) \(x=-1\) and \(x=0\). (Why should the answer be the same as the answer to the previous problem?) (b) \(x=0\) and \(x=1\).

The velocity of an object is given by \(3 \sin (\pi t)\). (a) What is the object's speed as a function of time? (b) What is the object's net displacement from \(t=0\) to \(t=2 ?\) (c) How far has the object traveled from \(t=0\) to \(t=2 ?\) (d) What is the object's average velocity on \([0,2] ?\) (e) What is the object's average speed on \([0,2]\) ?

Aimee and Alexandra spent Friday afternoon eating hot cinnamon hearts. If they gobbled hearts at a rate of \(1.5 t+\sqrt{t}\) hearts per minute, then how many hearts did they consume between time \(t=0\) and \(t=9, t\) given in minutes?

Suppose the temperature of an object is changing at a rate of \(r(t)=-2 e^{-t}\) degrees Celsius per hour, where \(t\) is given in hours. (a) Is the object heating, or cooling? (b) Between time \(t=0\) and \(t=1\), how much has the temperature changed? (c) Between \(t=1\) and \(t=2\), how much has the temperature changed? (d) If the object was 100 degrees Celsius at time \(t=0\), how hot is it at time \(t=1\) ?

A bicycle speedometer will give the average velocity of a bicyclist over the time period the bicycle is moving. By pressing a button the bicylist can reset the average velocity counter. Suppose a long-distance cyclist has averaged 14 miles per hour for the first two hours of her trip. She resets the average velocity counter. For the next four hours her average velocity is 18 miles per hour. (a) What is the cyclist's average velocity for the six-hour trip? (b) How far has she traveled?

Let \(f(x)=\int_{1}^{x} \frac{1}{t} d t, x>0\). (a) Find \(f(1), f(5), f(10)\), and \(f(1 / 2)\). (b) What is an alternative formula for \(f(x)\) ? (c) Often mathematicians de ne the natural logarithm by \(\ln x=\int_{1}^{x} \frac{1}{t} d t\) for \(x>0\). Suppose this was the definition you had been given. Use the Fundamental Theorem of Calculus to show that \(\ln x\) is increasing and concave down for \(x>0\).

Find the value of \(x>1\) such that the area under the graph of \(1 / t\) from 1 to \(x\) is 1 .

The temperature of a hotplate of radius 5 inches varies with the distance from the center of the plate. For the area within 2 inches of the center the average temperature is 100 degrees. For the area between 2 and 5 inches from the center the average temperature is 80 degrees. What is the average temperature of the plate?

The rate of change of water level in a tank is given by \(r(t)=2 \sin \left(\frac{\pi}{4} t\right)\) gallons per hour, where \(t\) is measured in hours. At time \(t=0\) there are 30 gallons of water in the tank. (a) Between time \(t=0\) and \(t=8\), when will the water level in the tank be the highest? (b) What is the maximum amount of water that will ever be in the tank? (c) What is the minimum amount of water that will ever be in the tank? (d) Is the amount of water added to the tank between \(t=0\) and \(t=1\) less than, greater than, or equal to the amount lost between \(t=4\) and \(t=5\) ? (Try to answer without doing any computations.)

Find the average value of \(|\sin (3 t)|\) on \([0,2 \pi] .\) Explain your reasoning.

Let \(g(x)=\int_{0}^{x} e^{-t^{2}} d t\). (a) Where is \(g(x)\) zero? Positive? Negative? (b) Where is \(g(x)\) increasing? Decreasing? (c) Where is \(g(x)\) concave up? Concave down? (d) Is \(g(x)\) even, odd, or neither? (e) Although we cannot find an antiderivative for \(e^{-t^{2}}\), we are able to get a lot of information about \(g(x) .\) Sketch \(g(x)\). (f) Using a computer or programmable calculator, approximate \(g(1), g(-1)\), and \(g(2) .\) Go back and look at all your answers to this question; make sure that they are consistent with one another.

Let \(h(x)=\int_{0}^{x} \sin \left(t^{2}\right) d t\). (a) Graph \(\sin \left(t^{2}\right)\) on the domain \([-3,3]\). (b) On \((0, \infty)\), where is \(h(x)\) positive? Negative? (c) Is \(h(x)\) even, odd, or neither? Explain. (d) On \([0,3]\) where is \(h(x)\) increasing? Decreasing? Give exact answers. (e) What is the absolute maximum value of \(\sin \left(t^{2}\right)\) ? What is the absolute minimum value of \(\sin \left(t^{2}\right) ?\) (f) Where on \([0,3]\) does \(h\) attain its maximum and minimum values? Will your answers change if the domain is \((0, \infty)\) ? If the domain is \((-\infty, \infty)\) ? (g) Numerically approximate the maximum value of \(h\) on \([0,3]\).

(a) Find \(\int_{-1}^{1}|x| d x\) using the area interpretation of the definite integral. (b) Show that \(\frac{\left|x^{2}\right|}{2}\) is not an antiderivative of \(|x|\) on \([-1,1]\) by showing that applying the Fundamental Theorem as if it were, gives the wrong answer. (c) Find an antiderivative of \(|x|\) on \([-1,0)\). Find an antiderivative of \(|x|\) on \((0,1]\).

Find the following two definite integrals without using the Fundamental Theorem of Calculus. Instead, use the area interpretation of the definite integral. (a) \(\int_{-3}^{7}(\pi+1) d x\) (b) \(\int_{-3}^{7}|-2 x-4| d x\)

Evaluate the following. (If you haven't done Problem 14 , do that first.) (a) \(\int_{-3}^{3}\left|x^{2}-4\right| d x\) (b) \(\int_{0}^{5}|(x+3)(x-1)| d x\)

Calculate the following definite integrals by calculating the limit of Riemann sums. You'll need to use the formulas provided. Check your answers using the Fundamental Theorem of Calculus. (a) \(\int_{0}^{5} x d x \quad 1+2+3+\cdots+n=\frac{n(n+1)}{2}\) (b) \(\int_{0}^{5} x^{2} d x \quad 1^{2}+2^{2}+3^{2}+\cdots n^{2}=\frac{n(n+1)(2 n+1)}{6}\) (c) \(\int_{0}^{2} x^{3} d x \quad 1^{3}+2^{3}+3^{3}+\cdots n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}\)

(a) Write a Riemann sum with 10 equal subdivisions that gives an overestimate for the area under \(\ln x\) on [1,6]. Write your answer in two ways, once with summation notation and one without summation notation (using \(+\cdots+\) ). State clearly and precisely the meaning of any notation used in your sum. (b) Consider the Riemann sum $$\sum_{k=0}^{49} \ln (3+k \cdot(5 / 50)) \cdot \frac{5}{50}$$ This is an underestimate for what integral? (The answer to this question is not unique.)

Put the following in ascending order, using \(<\) or \(=\) as appropriate. $$ \int_{1}^{2} \ln x d x \quad \int_{0.5}^{2} \ln x d x \quad \int_{1}^{2.5} \ln x d x $$

(a) Which of the following is an antiderivative of \(\ln x ?\) \(\begin{array}{llll}\text { i. } \frac{1}{x} & \text { ii. } x \ln x & \text { iii. } x \ln x-x & \text { iv. } \arctan (\ln x)\end{array}\) (b) Evaluate \(\int_{1}^{6} \ln x d x\).