Chapter 5

If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions. \(\int \sqrt{1+\sin ^{2}(x-1)} \sin (x-1) \cos (x-1) d x\) a. \(u=x-1,\) followed by \(v=\sin u,\) then by \(w=1+v^{2}\) b. \(u=\sin (x-1),\) followed by \(v=1+u^{2}\) c. \(u=1+\sin ^{2}(x-1)\)

Express the solutions of the initial value problems in terms of integrals. $$\frac{d y}{d x}=\sec x, \quad y(2)=3$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(63-72\). $$y=x^{4}-4 x^{2}+4 \text { and } y=x^{2}$$

Evaluate the integrals $$\int \frac{(2 r-1) \cos \sqrt{3(2 r-1)^{2}+6}}{\sqrt{3(2 r-1)^{2}+6}} d r$$

Express the solutions of the initial value problems in terms of integrals. $$\frac{d y}{d x}=\sqrt{1+x^{2}}, \quad y(1)=-2$$

Evaluate the integrals $$\int \frac{\sin \sqrt{\theta}}{\sqrt{\theta \cos ^{3} \sqrt{\theta}}} d \theta$$

Archimedes \((287-212\) B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times in the Western world, discovered that the area under a parabolic arch is two-thirds the base times the height. Sketch the parabolic arch \(y=h-\left(4 h / b^{2}\right) x^{2}\) \(-b / 2 \leq x \leq b / 2,\) assuming that \(h\) and \(b\) are positive. Then use calculus to find the area of the region enclosed between the arch and the \(x\) -axis.

What values of \(a\) and \(b\) maximize the value of $$ \int_{a}^{b}\left(x-x^{2}\right) d x ? $$ (Hint: Where is the integrand positive?)

Show that if \(k\) is a positive constant, then the area between the \(x\) -axis and one arch of the curve \(y=\sin k x\) is \(2 / k\).

The marginal cost of printing a poster when \(x\) posters have been printed is $$\frac{d c}{d x}=\frac{1}{2 \sqrt{x}}$$ dollars. Find \(c(100)-c(1),\) the cost of printing posters \(2-100\).

Find the areas of the regions enclosed by the lines and curves in Exercises \(73-80\). $$x=2 y^{2}, \quad x=0, \quad \text { and } \quad y=3$$

Use the Max-Min Inequality to find upper and lower bounds for the value of $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Solve the initial value problems $$\frac{d s}{d t}=12 t\left(3 t^{2}-1\right)^{3}, \quad s(1)=3$$

Suppose that a company's marginal revenue from the manufacture and sale of eggbeaters is $$\frac{d r}{d x}=2-2 /(x+1)^{2}$$ where \(r\) is measured in thousands of dollars and \(x\) in thousands of units. How much money should the company expect from a production run of \(x=3\) thousand eggbeaters? To find out, integrate the marginal revenue from \(x=0\) to \(x=3\).

Find the areas of the regions enclosed by the lines and curves in Exercises \(73-80\). $$x=y^{2} \quad \text { and } \quad x=y+2$$

Solve the initial value problems $$\frac{d y}{d x}=4 x\left(x^{2}+8\right)^{-1 / 3}, \quad y(0)=0$$

The temperature \(T\left(^{\circ} \mathrm{F}\right)\) of a room at time \(t\) minutes is given by $$T=85-3 \sqrt{25-t} \text { for } 0 \leq t \leq 25$$. a. Find the room's temperature when \(t=0, t=16,\) and \(t=25\). b. Find the room's average temperature for \(0 \leq t \leq 25\).

Find the areas of the regions enclosed by the lines and curves in Exercises \(73-80\). $$y^{2}-4 x=4 \quad \text { and } \quad 4 x-y=16$$

Show that the value of \(\int_{0}^{1} \sin \left(x^{2}\right) d x\) cannot possibly be 2

Solve the initial value problems $$\frac{d s}{d t}=8 \sin ^{2}\left(t+\frac{\pi}{12}\right), \quad s(0)=8$$

The height \(H\) (ft) of a palm tree after growing for \(t\) years is given by $$H=\sqrt{t+1}+5 t^{1 / 3} \quad \text { for } \quad 0 \leq t \leq 8$$. a. Find the tree's height when \(t=0, t=4,\) and \(t=8\). b. Find the tree's average height for \(0 \leq t \leq 8\).

Solve the initial value problems $$\frac{d r}{d \theta}=3 \cos ^{2}\left(\frac{\pi}{4}-\theta\right), \quad r(0)=\frac{\pi}{8}$$

Suppose that \(\int_{1}^{x} f(t) d t=x^{2}-2 x+1 .\) Find \(f(x)\).

Solve the initial value problems $$\frac{d^{2} s}{d t^{2}}=-4 \sin \left(2 t-\frac{\pi}{2}\right), \quad s^{\prime}(0)=100, \quad s(0)=0$$

Find \(f(4)\) if \(\int_{0}^{x} f(t) d t=x \cos \pi x\).

Integrals of nonpositive functions Show that if \(f\) is integrable then $$ f(x) \leq 0 \quad \text { on } \quad[a, b] \quad \Rightarrow \quad \int_{a}^{b} f(x) d x \leq 0 $$

Solve the initial value problems $$\frac{d^{2} y}{d x^{2}}=4 \sec ^{2} 2 x \tan 2 x, \quad y^{\prime}(0)=4, \quad y(0)=-1$$

Find the linearization of $$f(x)=2-\int_{2}^{x+1} \frac{9}{1+t} d t$$ at \(x=1\).

Use the inequality \(\sin x \leq x,\) which holds for \(x \geq 0,\) to find an upper bound for the value of \(\int_{0}^{1} \sin x d x\)

The velocity of a particle moving back and forth on a line is \(v=d s / d t=6 \sin 2 t \mathrm{m} / \mathrm{sec}\) for all \(t .\) If \(s=0\) when \(t=0,\) find the value of \(s\) when \(t=\pi / 2\) sec.

Find the linearization of$$g(x)=3+\int_{1}^{x^{2}} \sec (t-1) d t$$at \(x=-1\).

The inequality sec \(x \geq 1+\left(x^{2} / 2\right)\) holds on \((-\pi / 2, \pi / 2) .\) Use it to find a lower bound for the value of \(\int_{0}^{1} \sec x d x\)

The acceleration of a particle moving back and forth on a line is \(a=d^{2} s / d t^{2}=\pi^{2} \cos \pi t \mathrm{m} / \mathrm{sec}^{2}\) for all \(t .\) If \(s=0\) and \(v=\) \(8 \mathrm{m} / \mathrm{sec}\) when \(t=0,\) find \(s\) when \(t=1 \mathrm{sec}\)

Suppose that \(f\) has a positive derivative for all values of \(x\) and that \(f(1)=0 .\) Which of the following statements must be true of the function $$g(x)=\int_{0}^{x} f(t) d t ?$$. Give reasons for your answers. a. \(g\) is a differentiable function of \(x\). b. \(g\) is a continuous function of \(x\). c. The graph of \(g\) has a horizontal tangent at \(x=1\). d. \(g\) has a local maximum at \(x=1\). e. \(g\) has a local minimum at \(x=1\). f. The graph of \(g\) has an inflection point at \(x=1\). g. The graph of \(d g / d x\) crosses the \(x\) -axis at \(x=1\).

Find the areas of the regions enclosed by the curves in Exercises \(81-84\). $$4 x^{2}+y=4 \text { and } x^{4}-y=1$$

It would be nice if average values of integrable functions obeyed the following rules on an interval \([a, b]\) a. \(\operatorname{av}(f+g)=\operatorname{av}(f)+\operatorname{av}(g)\) b. \(\operatorname{av}(k f)=k \operatorname{av}(f) \quad\) (any number \(k\) ) c. \(\operatorname{av}(f) \leq \operatorname{ar}(g)\) if \(f(x) \leq g(x)\) on \([a, b]\) Do these rules ever hold? Give reasons for your answers.

Another proof of the Evaluation Theorem a. Let \(a=x_{0}

It would be nice if average values of integrable functions obeyed the following rules on an interval \([a, b]\) a. \(\operatorname{av}(f+g)=\operatorname{av}(f)+\operatorname{av}(g)\) b. \(\operatorname{av}(k f)=k \operatorname{av}(f) \quad\) (any number \(k\) ) c. \(\operatorname{av}(f) \leq \operatorname{av}(g)\) if \(f(x)=g(x)\) on \([a, b]\) Do these rules ever hold? Give reasons for your answers.

Find the areas of the regions enclosed by the curves in Exercises \(81-84\). $$x+4 y^{2}=4 \quad \text { and } \quad x+y^{4}=1, \quad \text { for } \quad x \geq 0$$

Find \(\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{d t}{\sqrt{t}}\).

Let \(F(x)=\int_{a}^{x} f(t) d t\) for the specified function \(f\) and interval \([a, b] .\) Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions \(f\) and \(F\) together over \([a, b]\) b. Solve the equation \(F^{\prime}(x)=0 .\) What can you see to be true about the graphs of \(f\) and \(F\) at points where \(F^{\prime}(x)=0\) ? Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function \(F\) increasing and decreasing? What is true about \(f\) over those intervals? d. Calculate the derivative \(f^{\prime}\) and plot it together with \(F .\) What can you see to be true about the graph of \(F\) at points where \(f^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer. $$f(x)=\sin 2 x \cos \frac{x}{3}, \quad[0,2 \pi]$$

If you average \(30 \mathrm{mi} / \mathrm{h}\) on a \(150-\mathrm{mi}\) trip and then return over the same 150 mi at the rate of \(50 \mathrm{mi} / \mathrm{h}\), what is your average speed for the trip? Give reasons for your answer.

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and f. Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F\). b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrema and the points of inflection of \(F .\) d. Using the information from parts (a)-(c), draw a rough handsketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch. $$a=1, \quad u(x)=x^{2}, \quad f(x)=\sqrt{1-x^{2}}$$

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and f. Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F\). b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrema and the points of inflection of \(F .\) d. Using the information from parts (a)-(c), draw a rough handsketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch. $$a=0, \quad u(x)=x^{2}, \quad f(x)=\sqrt{1-x^{2}}$$

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and f. Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F\). b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrema and the points of inflection of \(F .\) d. Using the information from parts (a)-(c), draw a rough handsketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch. $$a=0, \quad u(x)=1-x, \quad f(x)=x^{2}-2 x-3$$

If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals in Exercises \(89-94 .\) Use \(n=4,10,20,\) and 50 subintervals of equal length in each case. $$\int_{-\infty}^{\pi} \cos x d x=0$$

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and f. Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F\). b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrema and the points of inflection of \(F .\) d. Using the information from parts (a)-(c), draw a rough handsketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch. $$a=0, \quad u(x)=1-x^{2}, \quad f(x)=x^{2}-2 x-3$$

Assume that \(f\) is continuous and \(u(x)\) is twice-differentiable. Calculate \(\frac{d}{d x} \int_{a}^{u(x)} f(t) d t\) and check your answer using a CAS.

Assume that \(f\) is continuous and \(u(x)\) is twice-differentiable. Calculate \(\frac{d^{2}}{d x^{2}} \int_{a}^{u(x)} f(t) d t\) and check your answer using a CAS.

Find the area of the region in the first quadrant bounded by the line \(y=x,\) the line \(x=2,\) the curve \(y=1 / x^{2},\) and the \(x\) -axis.