Chapter 10

Use your technology to find a fourth degree polynomial close to the data in the table below taken from the graph of average solar intensity at Eugene, Oregon in Figure 9.1 .3 on page 405 . $$ \begin{array}{lrrrrrrr} \text { Day } & 1 & 60 & 120 & 180 & 240 & 300 & 360 \\ \text { Day minus 180 } & -179 & -120 & -60 & 0 & 60 & 120 & 180 \\ \text { Solar intensity, kw-hr/m }^{2} & 1.0 & 2.0 & 4.5 & 6.7 & 5.6 & 1.8 & 1.0 \end{array} $$ Your technology will object that the equations to compute the coefficients of a fourth degree polynomial to the Day - Solar Intensity data are ill conditioned (subject to roundoff error); the equations to fit Day minus 180 - Solar Intensity are OK. Therefore, fit a fourth degree polynomial to the Day minus 180 - Solar Intensity data, and interpret the polynomial according for plotting the data and for integration. You should get $$ \begin{array}{c} P(t)=8.733910^{-9}(t-180)^{4}-1.222510^{-7}(t-180)^{3}-4.558710^{-4}(t-180)^{2} \\\ +3.414910^{-3}(t-180)+6.63072 \end{array} $$ a. Plot the data and a graph of your polynomial. b. Compute the integral of \(\int_{0}^{365} P_{2}(t) d t\). c. Compare your answers to the estimate of 1324 computed using the trapezoid rule on 12 intervals of length 30 and one interval of length \(5,\) in Exercise 9.1 .3 on page 404 .

Check by differentiation the validity of the indefinite integral formulas: a. \(\int \frac{1}{t} \mathrm{dt}=\ln \mathrm{t}+\mathrm{C}\) b. \(\int[\mathrm{U}(\mathrm{t})]^{\mathrm{n}} \mathrm{U}^{\prime}(\mathrm{t}) \mathrm{dt}=\frac{\mathrm{U}(\mathrm{t})^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{C}\) c. \(\int \mathrm{e}^{\mathrm{kt}} \mathrm{dt}=\frac{1}{\mathrm{k}} \mathrm{e}^{\mathrm{kt}}+\mathrm{C}\)

Suppose \(P(t)\) is the size of a population at time \(t, P(0)=5000,\) and \(P^{\prime}(t)=0\) for all \(t\). What is \(P(100) ?\) What is \(P(10000000000) ?\)

Compute \(\int\left(1+t^{4}\right)^{3} d t\).

Evaluate the integrals. a. \(\int_{1}^{3} t^{2} d t\) b. \(\int_{0}^{2} t^{3} d t\) c. \(\int_{0}^{2} e^{t} d t\) d. \(\int_{1}^{3} \frac{1}{t} d t\) e. \(\int_{1}^{3}(1+t)^{2} d t\) f. \(\int_{1}^{3} 5 t d t\) g. \(\int_{0}^{2} t+5 d t\) $$ \text { h. } \int_{0}^{2} e^{2 t} d t \quad \text { i. } \quad \int_{0}^{3} e^{-t} d t \quad \text { j. } \quad \int_{1}^{3}\left(1+t^{2}\right) d t $$

Suppose a mold colony is growing in a nutrient solution and that on day zero the area was \(0.5 \mathrm{~cm}^{2}\) and for every time, \(t \geq 0\), the instantaneous rate of growth of the area of the colony is \(2 t \mathrm{~cm}^{2}\) per day. Let \(P(t)\) be the colony area at time \(t\). a. Argue that for every time, \(t, P^{\prime}(t)=2 t\). b. Show that for \(Q(t)=t^{2}, Q^{\prime}(t)=2 t\). Then \(P^{\prime}(t)=Q^{\prime}(t)\). c. From the Parallel Graph Theorem, it follows that there is a constant \(C\) such that \(P(t)=Q(t)+C\) d. Use \(P(0)=0.5\) to evaluate \(C\). e. What is the area of the mold colony on day \(8 ?\)

Which of the two indefinite integrals $$ \int \frac{1}{1+t^{2}} t d t \quad \text { or } \int \frac{1}{1+t^{2}} d t $$ is \(\ln \left(1+t^{2}\right)^{0.5}+C ?\) Explain your answer. Note: The other indefinite integral is arctan \(t+C\).

Suppose the rate of glucose production in a corn plant is proportional to sunlight intensity and can be approximated by $$ R(t)=K(t+7)^{2}(t-7)^{2}=K\left(t^{4}-98 t^{2}+2401\right) \quad-7 \leq t \leq 7 $$ Time is measured so that sunrise is at -7 hours, the sun is at its zenith at 0 hours and sets at 7 hours. The quantity \(Q(x)\) of glucose produced during the period \([-7, x]\) is $$ Q(x)=\int_{-7}^{x} R(t) d t=\int_{-7}^{x} K\left(t^{4}-98 t^{2}+2401\right) d t=K \int_{-7}^{x}\left(t^{4}-98 t^{2}+2401\right) d t $$ By the Fundamental Theorem of Calculus, $$ Q^{\prime}(x)=K\left(x^{4}-98 x^{2}+2401\right) $$ a. Sketch the graph of \(R(t)\). At what time is the sun most intense? b. Show that if \(U(x)=\frac{x^{5}}{5}\) then \(U^{\prime}(x)=x^{4}\). c. Find an example of a function, \(V(x)\) such that \(V^{\prime}(x)=-98 x^{2}\). d. Find an example of a function, \(W(x)\) such that \(W^{\prime}(x)=2401\). e. Let \(G(x)=K\left(\frac{x^{5}}{5}-\frac{98}{3} x^{3}+2401 x\right)\). Show that \(G^{\prime}(x)=Q^{\prime}(x)\). f. Conclude that there is a number, \(C\), such that $$ Q(x)=G(x)+C=K\left[\frac{x^{5}}{5}-\frac{98}{3} x^{3}+2401 x\right]+C $$ a. Sketch the graph of \(R(t)\). At what time is the sun most intense? b. Show that if \(U(x)=\frac{x^{5}}{5}\) then \(U^{\prime}(x)=x^{4}\). c. Find an example of a function, \(V(x)\) such that \(V^{\prime}(x)=-98 x^{2}\). d. Find an example of a function, \(W(x)\) such that \(W^{\prime}(x)=2401\). e. Let \(G(x)=K\left(\frac{x^{5}}{5}-\frac{98}{3} x^{3}+2401 x\right)\). Show that \(G^{\prime}(x)=Q^{\prime}(x)\). f. Conclude that there is a number, \(C,\) such that $$ Q(x)=G(x)+C=K\left[\frac{x^{5}}{5}-\frac{98}{3} x^{3}+2401 x\right]+C $$ g. Why is \(Q(-7)=0\). h. Evaluate \(C\). i. Compute \(Q(7),\) the amount of glucose produced during the day.

Let \(\mathrm{G}(\mathrm{x})\) be the area of the region bounded by the graphs of \(y=\frac{1}{1+t^{2}}, y=0,\) \(t=1\) and \(t=x .\) a. Compute approximate values for \(y(1.0), y(1.5), y(2.0), y(2.5),\) and \(y(3.0)\). b. Compute approximate values for \(G(1.0), G(1.5), G(2.0), G(2.5),\) and \(G(3.0)\). c. Sketch the graph of \(y=\frac{1}{1+t^{2}}\) on [0,3] . d. Sketch the graph of \(G\) on [1,3] . e. Estimate \(G^{\prime}(2)\).

Compute the following integrals and antiderivatives. For the definite integrals, draw a region in the plane whose area is computed by the integral. If you solve the integral by a substitution, \(u(t)=,\) then identify in writing \(u(t)\) and \(u^{\prime}(t)\) a. \(\int_{0}^{1} t^{4} d t\) b. \(\int_{0}^{1} t^{499} d t\) c. \(\int_{0}^{1} t^{1 / 2} d t\) d. \(\int_{0}^{2} e^{x} d x\) e. \(\int_{0}^{\pi} \cos z d z\) \(f . \quad \int_{2}^{6} \frac{1}{y} d y\) \(g . \quad \int_{12}^{36} \frac{1}{t} d t\) h. \(\int t^{-1 / 2} d t\) i. \(\int(\sin t+\cos t) d t\) \(j . \int \sqrt{t} d t\) k. \(\int\left(5 t^{4}+3 t^{2}+5\right) d t\) l. \(\int \frac{w^{2}+w+1}{w} d w\)

a. Let \(y=\arctan x\) so that \(x=\tan y(x)\). Show that $$ \sec ^{2} y(x)=1+x^{2} $$ b. Use \(x=\tan y(x)\) and the chain rule, \([G(u(x))]^{\prime}=G^{\prime}(u(x)) u^{\prime}(x),\) to conclude that $$ 1=\left(\sec ^{2} y(x)\right) y^{\prime}(x) \quad \text { and } \quad y^{\prime}(x)=\frac{1}{1+x^{2}} $$ c. Use the result of b. to show that $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x=\frac{\pi}{4} $$

At 1: 00 a.m. an oil pipe line bursts and starts releasing oil into a lake at the rate of 2 cubic meters per hour. At 2: 00 a.m., a second oil pipe line bursts and also starts releasing oil into the lake at the rate of 3 cubic meters per hour. a. How much oil is in the lake at 1: 00,1: 30,2: 00,2: 30,3: 00,\(3: 30,\) and \(4: 00 ?\) b. Let \(\mathrm{T}(\mathrm{x})\) be the total amount of oil in the lake at time \(x\). Draw a graph of \(T\). c. Write equations describing the total amount of oil, \(T(x),\) in the lake for each time \(x\) between 1:00 a.m. and 4: 00 a.m. d. Compute \(T^{\prime}\).

Compute the following antiderivatives or integrals. If you solve the integral by a substitution, \(u(t)=,\) then identify in writing \(u(t)\) and \(u^{\prime}(t)\) a. \(\int e^{2 t} d t\) b. \(\int \sin (2 t) 2 d t\) c. \(\int \frac{1}{\sqrt{t}} d t\) d. \(\int 3 z^{-1} d z\) e. \(\int(3 \cos z+4 \sin z) d z\) f. \(\int\left(\pi^{2}+e^{2}\right) d z\) g. \(\int \frac{x^{2}}{\sqrt{x}} d x\) h. \(\int \frac{x^{2}+1}{x} d x\) i. \(\int\left(x+\frac{1}{x}\right)^{2} d x\) \(j . \int \frac{1}{t+1} d t\) k. \(\int\left(1+t^{2}\right)^{3} 2 t d t\) l. \(\int \frac{1}{z^{5}} d z\)

Equal quantities of gaseous hydrogen and iodine are mixed resulting in the reaction $$ \mathrm{H}_{2}+\mathrm{I}_{2} \longrightarrow 2 \mathrm{HI} $$ which runs until \(I_{2}\) is exhausted \(\left(H_{2}\right.\) is also exhausted). The rate at which \(I_{2}\) disappears is \(\frac{0.2}{(t+1)^{2}}\) \(\mathrm{gm} / \mathrm{sec} .\) How much \(I_{2}\) was initially introduced into the mixture? a. Sketch the graph of the reaction rate, \(r(t)=\frac{0.2}{(t+1)^{2}}\). b. Approximately how much \(I_{2}\) combined with \(H_{2}\) during the first second? c. Approximately how much \(I_{2}\) combined with \(H_{2}\) during the second second? d. Let \(Q(x)\) be the amount of \(I_{2}\) that combines with \(H_{2}\) during time 0 to \(x\) seconds. Write an integral that is \(Q(x)\) e. What is \(Q^{\prime}(x) ?\) f. Compute \(W^{\prime}(x)\) for \(W(x)=\frac{-0.2}{1+x}\). g. Show that there is a number, \(C,\) for which \(Q(x)=W(x)+C\). h. Show that \(C=0.2\) so that \(Q(x)=0.2-\frac{0.2}{1+x}\). i. How much \(I_{2}\) combined with \(H_{2}\) during the first second? j. How much \(I_{2}\) combined with \(H_{2}\) during the first 100 seconds? k. How much \(I_{2}\) combined with \(H_{2} ?\)

Compute the following antiderivatives. If you solve the integral by a substitution, \(u(t)=\), then identify in writing \(u(t)\) and \(u^{\prime}(t)\). a. \(\int \sin ^{4}(t) \cos t d t\) b. \(\int t\left(1+t^{2}\right)^{3} d t\) c. \(\int \frac{1}{(z+1)^{3}} d z\) d. \(\int \frac{1}{4 t+1} d t\) e. \(\int t\left(1+t^{4}\right)^{3} d t\) \(f . \quad \int \frac{1}{3 z+1} d z\) \(g . \quad \int \frac{\sin t}{\cos t} d t\) h. \(\int(1+x)^{3} d x\) i. \(\int \frac{z}{z^{2}+1} d z\) \(j \cdot \int\left(1+t^{2}\right)^{3} t^{-1} d t\) k. \(\int e^{2+z} d z\) l. \(\int_{0}^{\pi} \sin (\pi+x) d x\) \(m . \int \sin (4 t) d t\) n. \(\int(\ln x) \frac{1}{x} d x\) o. \(\int e^{-z^{2}} z d z\)

Let \(F(t)=[t]\) where \([t]\) is the greatest integer less than or equal to \(t\). For example, \([\pi]=3,\) and \([2]=2 .\) Let \(G(x)=\int_{0}^{x} f(t) d t\) for \(0 \leq x \leq 4\) a. Sketch the graph of \(f\) for \(0 \leq t \leq 5\) b. Sketch the graph of \(G\). c. Sketch the graph of \(G^{\prime}\). d. You should find that \(G^{\prime} \neq f .\) Does this contradict the Fundamental Theorem of Calculus?

a. (a) Compute \([x \ln x-x]^{\prime}\) (b) Compute \(\int_{1}^{e} \ln x d x\) b. Note: \(\sin ^{2} \theta=\frac{1-\cos 2 \theta}{2}\) Compute \(\int_{0}^{\pi} \sin ^{2} \theta d \theta\) c. (a) Show that \(\frac{1}{t}+\frac{1}{1-t}=\frac{1}{t(1-t)}\) (b) Compute \(\int \frac{1}{t(1-t)} d t\) d. \(\quad\) (a) Believe: \([\arcsin x]^{\prime}=\frac{1}{\sqrt{1-x^{2}}}\) (b) Compute \(\int_{0}^{1 / 2} \frac{1}{\sqrt{1-x^{2}}} d x\)

Solve using integration by parts, \(\int u(x) v^{\prime}(x) d x=u(x) v(x)-\int v(x) u^{\prime}(x) d x\) (or \(\left.\int u d v=u v-\int v d u\right)\) a. \(\int x e^{x} d x\) b. \(\int x \ln x d x\) c. \(\int x \sin x d x\) d. \(\int x^{2} e^{x} d x\) e. \(\int x e^{2 x} d x\) f. \(\int \ln x \cdot 1 d x\) g. \(\int x \cos x d x\) h. \(\int x^{3} e^{x^{2}} d x\)

Exercise 10.5 .10 a. Use integration by parts on \(\int e^{x} \sin x d x,\) with \(u(x)=e^{x}\) and \(v^{\prime}(x)=\sin x,\) to show that $$ \int e^{x} \sin x d x=-e^{x} \cos x+\int e^{x} \cos x d x $$ Use a second step integration by parts on \(\int e^{x} \cos x d x\) to show that $$ \int e^{x} \cos x d x=e^{x} \sin x-\int e^{x} \sin x d x $$ Combine the previous two equations to show that $$ \int e^{x} \sin x d x=\frac{1}{2} e^{x}(\sin x-\cos x)+C $$ b. Do two steps of integration by parts on \(\int e^{x} \cos x d x \quad\) and show that \(\quad \int e^{x} \cos x d x=\frac{1}{2} e^{x}(\sin x+\cos x)+C\) c. Do two steps of integration by parts on \(\int(\sin x) e^{-x} d x \quad\) and show that \(\quad \int_{0}^{\pi} e^{-x} \sin x d x=\frac{1}{2}\left(e^{-\pi}+1\right)\) d. Clever! Note that \(\int e^{\sqrt{x}} d x=\int 2 \sqrt{x} e^{\sqrt{x}} \frac{1}{2 \sqrt{x}} d x\). Let \(u=2 \sqrt{x}\) and \(v^{\prime}=e^{\sqrt{x}} \frac{1}{2 \sqrt{x}}\) and show that $$ \int e^{\sqrt{x}} d x=2 \sqrt{x} e^{\sqrt{x}}-2 e^{\sqrt{x}}+C $$