Chapter 4

Compute \(\int_{0}^{4} f(x) d x\). $$f(x)=\left\\{\begin{array}{ll} 2 & \text { if } x \leq 2 \\ 3 x & \text { if } x>2 \end{array}\right.$$

Determine the position function if the velocity function is \(v(t)=3 e^{-t}-2\) and the initial position is \(s(0)=0\)

A function \(f\) is said to be even if \(f(-x)=f(x)\) for all \(x\) A function \(f\) is said to be odd if \(f(-x)=-f(x) .\) Suppose that \(f\) is continuous for all \(x\). Show that if \(f\) is even, then \(\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x .\) Also, if \(f\) is odd, show that \(\int_{-a}^{a} f(x) d x=0\)

Explain why Simpson's Rule can't be used to approximate \(\int_{0}^{\pi} \frac{\sin x}{x} d x .\) Find \(L=\lim _{x \rightarrow 0} \frac{\sin x}{x}\) and argue that if \(f(x)=\left\\{\begin{array}{ll}\frac{\sin x}{x} & \text { if } x \neq 0 \\ L & \text { if } x=0\end{array} \text { then } \int_{0}^{\pi} f(x) d x=\int_{0}^{\pi} \frac{\sin x}{x} d x\right.\) Use an appropriate numerical method to conjecture that \(\int_{0}^{\pi} \frac{\sin x}{x} d x \approx 1.18\left(\frac{\pi}{2}\right)\)

Find the position function \(s(t)\) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. $$a(t)=4-t, v(0)=8, s(0)=0$$

Suppose that, for a particular population of organisms, the birth rate is given by \(b(t)=410-0.3 t\) organisms per month and the death rate is given by \(a(t)=390+0.2 t\) organisms per month. Explain why \(\int_{0}^{12}[b(t)-a(t)] d t\) represents the net change in population in the first 12 months. Determine for which values of \(t\) it is true that \(b(t)>a(t) .\) At which times is the population increasing? Decreasing? Determine the time at which the population reaches a maximum.

Determine the position function if the acceleration function is \(a(t)=3 \sin t+1,\) the initial velocity is \(v(0)=0\) and the initial position is \(s(0)=4\)

Assume that \(f\) is periodic with period \(T\); that is \(_{11}\) \(f(x+T)=f(x)\) for all \(x\). Show that \(\int_{0}^{T} f(x) d x=\int_{a}^{a+T} f(x) d x\) for any real number \(a\). (Hint: First, work with \(0 \leq a \leq T\).)

Find the position function \(s(t)\) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. $$a(t)=16-t^{2}, v(0)=0, s(0)=30$$

Suppose that, for a particular population of organisms, the birth rate is given by \(b(t)=400-3\) sin \(t\) organisms per month and the death rate is given by \(a(t)=390+t\) organisms per month. Explain why \(\int_{0}^{12}[b(t)-a(t)] d t\) represents the net change in population in the first 12 months. Graphically determine for which values of \(t\) it is true that \(b(t)>a(t) .\) At which times is the population increasing? Decreasing? Estimate the time at which the population reaches a maximum.

Determine the position function if the acceleration function is \(a(t)=t^{2}+1,\) the initial velocity is \(v(0)=4\) and the initial position is \(s(0)=0\)

For the integral \(I=\int_{0}^{10} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{10-x}} d x,\) use a substitution to show that \(I=\int_{0}^{10} \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} d x .\) Use these two representations of 1 to evaluate \(I\)

In most of the calculations that you have done, it is true that the Trapezoidal Rule and Midpoint Rule are on opposite sides of the exact integral (i.e., one is too large, the other too small). Also, you may have noticed that the Trapezoidal Rule tends to be about twice as far from the exact value as the Midpoint Rule.Given this, explain why the linear combination \(\frac{1}{3} T_{n}+\frac{2}{3} M_{n}\) should give a good estimate of the integral. (Here, \(T_{n}\) represents the Trapezoidal Rule approximation using \(n\) partitions and \(M_{n}\) the corresponding Midpoint Rule approximation.)

For a particular ideal gas at constant temperature, pressure \(P\) and volume \(V\) are related by \(P V=10 .\) The work required to increase the volume from \(V=2\) to \(V=4\) is given by the integral \(\int_{2}^{4} P(V) d V .\) Estimate the value of this integral.

Suppose that a car can accelerate from 30 mph to 50 mph in 4 seconds. Assuming a constant acceleration, find the acceleration (in miles per second squared) of the car and find the distance traveled by the car during the 4 seconds.

Suppose that the temperature \(t\) months into the year is given by \(T(t)=64-24 \cos \frac{\pi}{6} t\) (degrees Fahrenheit). Estimate the average temperature over an entire year. Explain why this answer is obvious from the graph of \(T(t)\)

Suppose that a car can come to rest from 60 mph in 3 seconds. Assuming a constant (negative) acceleration, find the acceleration (in miles per second squared) of the car and find the distance traveled by the car during the 3 seconds (i.e., the stopping distance).

Generalize exercise 51 to \(I=\int_{0}^{d} \frac{f(x)}{f(x)+f(a-x)} d x\) for any positive, continuous function \(f\) and then quickly evaluate \(\int_{0}^{\pi / 2} \frac{\sin x}{\sin x+\cos x} d x\)

Find the average value of the function on the given interval. \(f(x)=x^{2}-1,[1,3]\)

Suppose that the average value of a function \(f(x)\) over the interval [0,2] is 5 and the average value of \(f(x)\) over the interval [2,6] is \(11 .\) Find the average value of \(f(x)\) over the interval [0,6].

Generalize the result of exercise 53 to \(\int_{2}^{4} \frac{f(9-x)}{f(9-x)+f(x+3)} d x\) for any positive, continuous function \(f\) on [2,4]

Find the average value of the function on the given interval. \(f(x)=x^{2}+2 x,[0,1]\)

Suppose that the average value of a function \(f(x)\) over an interval \([a, b]\) is \(v\) and the average value of \(f(x)\) over the interval [b, \(c]\) is \(w .\) Find the average value of \(f(x)\) over the interval \([a, c]\).

As in exercise \(54,\) evaluate \(\int_{0}^{2} \frac{f(x+4)}{f(x+4)+f(6-x)} d x\) for any positive, continuous function \(f\) on [0,2]

Find the average value of the function on the given interval. \(f(x)=2 x-2 x^{2},[0,1]\)

Use a geometric formula to compute the integral. $$\int_{0}^{2} 3 x d x$$

Use the substitution \(u=x^{1 / 6}\) to evaluate \(\int \frac{1}{x^{5 / 6}+x^{2 / 3}} d x\)

Find the average value of the function on the given interval. \(f(x)=x^{3}-3 x^{2}+2 x,[1,2]\)

Use a geometric formula to compute the integral. $$\int_{1}^{4} 2 x d x$$

Find the average value of the function on the given interval. \(f(x)=\cos x,[0, \pi / 2]\)

Use a geometric formula to compute the integral. $$\int_{0}^{2} \sqrt{4-x^{2}} d x$$

The following table shows the velocity of a falling object at different times. For each time interval, estimate the distance fallen and the acceleration. $$\begin{array}{|l|l|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 0.5 & 1.0 & 1.5 & 2.0 \\ \hline v(t)(\mathrm{f} t / \mathrm{s}) & -4.0 & -19.8 & -31.9 & -37.7 & -39.5 \\\ \hline \end{array}$$

Generalize exercises 56 and 57 to \(\int \frac{1}{x^{(p+1 / / q}+x^{p / q}} d x\) for positive integers \(p\) and \(q\)

Find the average value of the function on the given interval. \(f(x)=\sin x,[0, \pi / 2]\)

Use a geometric formula to compute the integral. $$\int_{-3}^{0} \sqrt{9-x^{2}} d x$$

The following table shows the velocity of a falling object at different times. For each time interval, estimate the distance fallen and the acceleration. $$\begin{array}{|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 1.0 & 2.0 & 3.0 & 4.0 \\ \hline v(t)(\mathrm{m} / \mathrm{s}) & 0.0 & -9.8 & -18.6 & -24.9 & -28.5 \\ \hline \end{array}$$

There are often multiple ways of computing an antiderivative. For \(\int \frac{1}{x \ln \sqrt{x}} d x,\) first use the substitution \(u=\ln \sqrt{x}\) to find the indefinite integral \(2 \ln |\ln \sqrt{x}|+c .\) Then rewrite \(\ln \sqrt{x}\) and use the substitution \(u=\ln x\) to find the indefinite integral 2 In \(|\ln x|+c .\) Show that these two answers are equivalent.

The table shows the temperature at different times of the day. Estimate the average temperature using (a) right-endpoint evaluation and (b) left-endpoint evaluation. Explain why the estimates are different. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { time } & 12: 00 & 3: 00 & 6: 00 & 9: 00 & 12: 00 & 3: 00 & 6: 00 & 9: 00 & 12: 00 \\ \hline \text { temperature } & 46 & 44 & 52 & 70 & 82 & 86 & 80 & 72 & 56 \\ \hline \end{array}$$

The following table shows the acceleration of a car moving in a straight line. If the car is traveling \(70 \mathrm{ft} / \mathrm{s}\) at time \(t=0\) estimate the speed and distance traveled at each time. $$\begin{array}{|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 0.5 & 1.0 & 1.5 & 2.0 \\ \hline a(t)\left(\mathrm{f} \downarrow \mathrm{s}^{2}\right) & -4.2 & 2.4 & 0.6 & -0.4 & 1.6 \\ \hline \end{array}$$

The following table shows the acceleration of a car moving in a straight line. If the car is traveling \(20 \mathrm{m} / \mathrm{s}\) at time \(t=0\) estimate the speed and distance traveled at each time. $$\begin{array}{|l|l|l|l|l|l|} \hline t(s) & 0 & 0.5 & 1.0 & 1.5 & 2.0 \\ \hline a(t)\left(\mathrm{m} / \mathrm{s}^{2}\right) & 0.6 & -2.2 & -4.5 & -1.2 & -0.3 \\ \hline \end{array}$$

Involve the just-in-time inventory discussed in the chapter introduction. For a business using just-in-time inventory, a delivery of \(Q\) items arrives just as the last item is shipped out. Suppose that items are shipped out at the constant rate of \(r\) items per day. If a delivery arrives at time \(0,\) show that \(f(t)=Q-r t\) gives the number of items in inventory for \(0 \leq t \leq \frac{Q}{r} .\) Find the average value of \(f\) on the interval \(\left[0, \frac{Q}{r}\right]\).

Find a function \(f(x)\) such that the point (1,2) is on the graph of \(y=f(x),\) the slope of the tangent line at (1,2) is 3 and \(f^{\prime \prime}(x)=x-1\)

Find each mistake in the following calculations and then show how to correctly do the substitution. Start with \(\int_{0}^{\pi} \cos ^{2} x d x=\int_{0}^{\pi} \cos x(\cos x) d x\) and then use the substitu- tion \(u=\sin x\) with \(d u=\cos x d x .\) Then $$\int_{0}^{\pi} \cos x(\cos x) d x=\int_{0}^{0} \sqrt{1-u^{2}} d u=0$$

Use the Fundamental Theorem of Calculus to find an antiderivative of \(\sin \sqrt{x^{2}+1}\)

Involve the just-in-time inventory discussed in the chapter introduction. The Economic Order Quantity (EOQ) model uses the assumptions in exercise 61 to determine the optimal quantity \(Q\) to order at any given time. Assume that \(D\) items are ordered annually, so that the number of shipments equals \(\frac{n}{Q}\). If \(C_{o}\) is the cost of placing an order and \(C_{c}\) is the annual cost for storing an item in inventory, then the total annual cost is given by \(f(Q)=C_{c} \frac{D}{Q}+C_{c} \frac{Q}{2} .\) Find the value of \(Q\) that minimizes the total cost. For the optimal order size, show that the total ordering cost \(C_{o} \frac{D}{Q}\) equals the total carrying cost (for storage) \(C_{c} \frac{Q}{2}\).

Find a function \(f(x)\) such that the point (-1,1) is on the graph of \(y=f(x),\) the slope of the tangent line at (-1,1) is 2 and \(f^{\prime \prime}(x)=6 x+4\)

For \(a>0,\) show that \(\int_{a}^{1} \frac{1}{x^{2}+1} d x=\int_{1}^{1 / a} \frac{1}{x^{2}+1} d x .\) Use this equality to derive an identity involving tan \(^{-1} \bar{x}\)

The number of items that consumers are willing to buy depends on the price of the item. Let \(p=D(q)\) represent the price (in dollars) at which \(q\) items can be sold. The integral \(\int_{0}^{Q} D(q) d q\) is interpreted as the total number of dollars that consumers would be willing to spend on \(Q\) items. If the price is fixed at \(P=D(Q)\) dollars, then the actual amount of money spent is \(P Q .\) The consumer surplus is defined by \(C S=\int_{0}^{Q} D(q) d q-P Q .\) Compute the consumer surplus for \(D(q)=150-2 q-3 q^{2}\) at \(Q=4\) and at \(Q=6 .\) What does the difference in \(C S\) values tell you about how many items to produce?

Involve the just-in-time inventory discussed in the chapter introduction. The EOQ model of exercise 62 can be modified to take into account noninstantaneous receipt. In this case, instead of a full delivery arriving at one instant, the delivery arrives at a rate of \(p\) items per day. Assume that a delivery of size \(Q\) starts at time \(0,\) with shipments out continuing at the rate of \(r\) items per day (assume that \(p>r\) ). Show that when the delivery is completed, the inventory equals \(Q(1-r / p) .\) From there, inventory drops at a steady rate of \(r\) items per day until no items are left. Show that the average inventory equals \(\frac{1}{2} Q(1-r / p)\) and find the order size \(Q\) that minimizes the total cost.

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int 2 x \cos x^{2} d x$$