Chapter 5

If a linear function passes through two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) what is the average value of the function on the interval from \(x_{1}\) to \(x_{2}\) ?

An average young female in the United States gains weight at the rate of \(14(x-10)^{-1 / 2}\) pounds per year, where \(x\) is her age \((10 \leq x \leq 19)\). Find the total weight gain from age 11 to \(19 .\)

On a hot summer afternoon, a city's electricity consumption is \(-3 t^{2}+18 t+10\) units per hour, where \(t\) is the number of hours after noon \((0 \leq t \leq 6) .\) Find the total consumption of electricity between the hours of 1 and \(5 \mathrm{p} \cdot \mathrm{m}\)

If the values of a function on an interval are always positive, can the average value of the function over that interval be negative?

An average young male in the United States gains weight at the rate of \(18(x-10)^{-1 / 2}\) pounds per year, where \(x\) is his age \((11 \leq x \leq 20)\). Find the total weight gain from age 11 to \(19 .\)

An average child of age \(x\) years grows at the rate of \(6 x^{-1 / 2}\) inches per year (for \(2 \leq x \leq 16\) ). Find the total height gain from age 4 to age \(9 .\)

If the values of a function on an interval are always greater than 7 , what can you say about the average value of the function on that interval?

A friend says that finding differentials is as easy as finding derivatives-you just multiply the derivative by \(d x\). Is your friend right?

After \(t\) hours of work, a bank clerk can process checks at the rate of \(r(t)\) checks per hour for the function \(r(t)\) given below. How many checks will the clerk process during the first three hours (time 0 to time 3 )? $$ r(t)=-t^{2}+90 t+5 $$

Can the average value of a function on an interval be larger than the maximum value of the function on that interval? Can it be smaller than the minimum value on that interval?

A friend says that if you can move numbers across the integral sign, you can do the same for variables since variables stand for numbers, and in this way you can always "fix" the differential \(d u\) to be what you want. Is your friend right?

After \(t\) hours of work, a bank clerk can process checks at the rate of \(r(t)\) checks per hour for the function \(r(t)\) given below. How many checks will the clerk process during the first three hours (time 0 to time 3 )? $$ r(t)=-t^{2}+60 t+9 $$

If the average value of \(f(x)\) on an interval is a number \(c\), what will be the average value of the function \(-f(x)\) on that interval?

Explain why the substitution \(u=x\) is useless for finding an integral by the substitution method. Give an example.

A company's marginal cost function is \(M C(x)\) (given below), where \(x\) is the number of units. Find the total cost of the first hundred units \((x=0\) to \(x=100)\) $$ M C(x)=6 e^{-0.02 x} $$

If two curves cross twice, you can find the area contained by them by evaluating one definite integral (integrating "upper minus lower"). What if the curves cross three times - how many integrations of "upper minus lower" would you need? What if the curves cross ten times?

A company's marginal cost function is \(M C(x)\) (given below), where \(x\) is the number of units. Find the total cost of the first hundred units \((x=0\) to \(x=100)\) $$ M C(x)=8 e^{-0.01 x} $$

What is wrong with the following use of the substitution \(u=x^{2} ?\) $$ \int e^{x^{2}} d x=\int e^{u} d u=e^{u}+C=e^{x^{2}}+C $$

Suppose that a company found its sales rate (in sales per day) if it did advertise, and also its (lower) sales rate if it did not advertise. If you integrated "upper minus lower" over a month, describe the meaning of the number that you would find.

BUSINESS: Money Stock Measure From 1964 to 2014 the money stock measure "M1" (currency, traveler's checks, demand deposits, and other checkable deposits) was growing at the rate of approximately \(94 e^{0.56 x}\) billion dollars per decade, where \(x\) is the number of decades since 1964 . Find the total increase in M1 from $$ 1964 \text { to } 2014 $$

What is wrong with the following use of the substitution \(u=x^{2}-1 ?\) \(\int \frac{1}{x^{2}-1} d x=\int \frac{1}{u} d u=\ln |u|+C=\ln \left|x^{2}-1\right|+C\)

Suppose that a company found its rate of revenue (dollars per day) and its (lower) rate of costs (also in dollars per day). If you integrated "upper minus lower" over a month, describe the meaning of the number that you would find.

BUSINESS: Money Stock Measure From 1984 to 2014 the money stock measure "M2" (M1 plus retail money market mutual funds, savings, and small time deposits) was growing at the rate of approximately \(1130 e^{0.54 x}\) billion dollars per decade, where \(x\) is the number of decades since 1984 . Find the total increase in M2 from 1984 to \(2014 .\)

Al and Betty carry out the substitution method for definite integrals slightly differently. Both use the same substitution to change the original " \(x^{\prime \prime}\) integral into \(\mathrm{a}^{\prime \prime} u^{\prime \prime}\) integral, which they then integrate. Then Al changes the " \(x^{\prime \prime}\) limits of integration into their corresponding " \(u^{\prime \prime}\) values, substitutes, and subtracts to get the final answer. Betty uses the substitution to return the integrand to its \({ }^{\prime \prime} x^{\prime \prime}\) form and uses the original limits of integration to get the final answer. Which way is better?

\(85-94 .\) The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int(x+1)(x-5)^{4} d x $$

GENERAL: Price Increase The price of a double-dip ice cream cone is increasing at the rate of \(15 e^{0.05 t}\) cents per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 . Find the total change in price between the years 2014 and 2024 .

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int(x-2)(x+4)^{5} d x $$

An automobile dealer estimates that the newest model car will sell at the rate of \(30 / t\) cars per month, where \(t\) is measured in months and \(t=1\) corresponds to the beginning of January. Find the number of cars that will be sold from the beginning of January to the beginning of May.

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int x(x-2)^{6} d x $$

World consumption of tin is running at the rate of \(342 e^{0.02 t}\) thousand metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 . Find the total consumption of tin from 2014 to 2024

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int x(x+4)^{7} d x $$

SOCIOLOGY: Marriages The marriage rate (marriages per year) in the United States has been declining recently, with about \(1.97 e^{-0.0102 t}\) million marriages per year, where \(t\) is the number of years since 2014 . Assuming that this rate continues, find the total number of marriages in the United States from 2014 to 2024

BEHAVIORAL SCIENCE: Learning A student can memorize words at the rate of \(6 e^{-t / 5}\) words per minute after \(t\) minutes. Find the total number of words that the student can memorize in the first 10 minutes.

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int \frac{x-4}{x-5} d x $$

BIOMEDICAL: Epidemics An epidemic is spreading at the rate of \(12 e^{0.2 t}\) new cases per day, where \(t\) is the number of days since the epidemic began. Find the total number of new cases in the first 10 days of the epidemic.

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int x \sqrt[3]{x-4} d x $$

GENERAL: Area a. Use your graphing calculator to find the area between 0 and 1 under the following curves: \(y=x, \quad y=x^{2}, \quad y=x^{3}, \quad\) and \(\quad y=x^{4}\) b. Based on your answers to part (a), conjecture a formula for the area under \(y=x^{n}\) between 0 and 1 for any value of \(n>0\) c. Prove your conjecture by evaluating an appropriate definite integral "by hand."

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int(x-1) \sqrt{x+2} d x $$

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int \frac{x}{\sqrt{x+2}} d x $$

An oral medication is absorbed into the bloodstream at the rate of \(5 e^{-0.04 t}\) milligrams per minute, where \(t\) is the number of minutes since the medication was taken. Find the total amount of medication absorbed within the first 30 minutes.

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int \frac{x}{\sqrt[3]{x+1}} d x $$

The rate of change of the volume of blood in the aorta \(t\) seconds after the beginning of the cardiac cycle is \(-k P_{0} e^{-m t}\) milliliters per second, where \(k, P_{0},\) and \(m\) are constants (depending, respectively, on the elasticity of the aorta, the initial aortic pressure, and various characteristics of the cardiac cycle). Find the total change in volume from time 0 to time \(T\) (the end of the cardiac cycle). (Your answer will involve the constants \(k, P_{0}, m,\) and \(\left.T .\right)\)

A dealer predicts that new cars will sell at the rate of \(8 x e^{-0.1 x}\) sales per week in week \(x\). Find the total sales in the first half year (week 0 to week 26 ).

A resort community swells at the rate of \(100 e^{0.4 \sqrt{x}}\) new arrivals per day on day \(x\) of its "high season." Find the total number of arrivals in the first two weeks (day 0 to day 14 ).

After \(t\) hours of work, a medical technician can carry out T-cell counts at the rate of \(2 t^{2} e^{-t / 4}\) tests per hour. How many tests will the technician process during the first eight hours (time 0 to time 8 )?

Suppose that you have a positive, increasing function and you approximate the area under it by a Riemann sum with left rectangles. Will the Riemann sum overestimate or underestimate the actual area? [Hint: Make a sketch.

Suppose that you have a positive, decreasing function and you approximate the area under it by a Riemann sum with left rectangles. Will the Riemann sum overestimate or underestimate the actual area? [Hint: Make a sketch.

Suppose that you have a positive, increasing, concave up function and you approximate the area under it by a Riemann sum with midpoint rectangles. Will the Riemann sum overestimate or underestimate the actual area? [Hint: Make a sketch.]

Suppose that you have a positive, increasing, concave down function and you approximate the area under it by a Riemann sum with midpoint rectangles. Will the Riemann sum overestimate or underestimate the actual area? [Hint: Make a sketch.

A friend says that definite and indefinite integrals are exactly the same except that one has numbers plugged in. A second friend disagrees, saying that the essential ideas are entirely different. Who is right?