Chapter 5

Evaluate the definite integral. \( \displaystyle \int^1_0 \frac{e^z + 1}{e^z + z} \, dz \)

A bacteria population is 4000 at time \( t = 0 \) and its rate of growth is \( 1000 \cdot 2^t \) bacteria per hour after \( t \) hours. What is the population after one hour?

Let \( f(x) = 0 \) if \( x \) is any rational number and \( f(x) = 1 \) if \( x \) is any irrational number. Show that \( f \) is not integrable on \( [0, 1] \).

Evaluate the definite integral. \( \displaystyle \int^{T/2}_0 \sin (2\pi t/T - \alpha) \, dt \)

The sine integral function $$ \displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt $$ is important in electrical engineering. [The integrand \( f(t) = (\sin t)/t \) is not defined when \( t = 0 \), but we know that its limit is 1 when \( t \to 0 \). So we defined \( f(0) = 1 \) and this makes \( f \) a continuous function everywhere.] (a) Draw the graph of \( \text{Si} \). (b) At what values of \( x \) does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation correct to one decimal place: $$ \displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1 $$

Let \( f(0) = 0 \) and \( f(x) = 1/x \) if \( 0 < x \le 1 \). Show that \( f \) is not integrable on \( [0, 1] \). [\( Hint: \) Show that the first term in the Riemann sum, \( f(x_i^*) \Delta x \), can be made arbitrarily large.]

Evaluate the definite integral. \( \displaystyle \int^1_0 \frac{dx}{(1 + \sqrt{x})^4} \)

Express the limit as a definite integral. \( \displaystyle \lim_{n \to \infty} \sum^{n}_{i = 1} \frac{i^4}{n^5} \) [\( Hint: \) Consider \( f(x) = x^4 \).]

Verify that \( f(x) = \sin \sqrt[3]{x} \) is an odd function and use that fact to show that $$ 0 \le \int^3_{-2} \sin \sqrt[3]{x} \, dx \le 1 $$

Express the limit as a definite integral. \( \displaystyle \lim_{n \to \infty} \frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2} \)

Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. \( y = \sqrt{2x + 1} \), \( 0 \le x \le 1 \)

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on \( [0, 1] \). \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \biggl( \frac{i^4}{n^5} + \frac{i}{n^2} \biggr) \)

Find \( \displaystyle \int^2_1 x^{-2} \,dx \). \( Hint: \) Choose \( x_i^* \) to be the geometric mean of \( x_{i - 1} \) and \( x_i \) (that is, \( x_i^* = \sqrt{x_{i - 1}x_i} \)) and use the identity $$ \frac{1}{m(m + 1)} = \frac{1}{m} - \frac{1}{m + 1} $$

Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. \( y = 2 \sin x - \sin 2x \), \( 0 \le x \le \pi \)

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on \( [0, 1] \). \( \displaystyle \lim_{n \to \infty} \frac{1}{n} \biggl( \sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + \cdots + \sqrt{\frac{n}{n}} \biggr) \)

Evaluate \( \displaystyle \int^2_{-2} (x + 3) \sqrt{4 - x^2} \,dx \) by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

Evaluate \( \displaystyle \int^1_0 x \sqrt{1 - x^4} \,dx \) by making a substitution and interpreting the resulting integral in terms of an area.

If \( f \) is continuous and \( g \) and \( h \) are differentiable functions, find a formula for $$ \displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt $$

A model for the basal metabolism rate, in kcal/h, of a young man is \( R(t) = 85 - 0.18 \cos(\pi t/12) \), where \( t \) is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, \( \displaystyle \int^{24}_0 R(t) \, dt \), over a 24-hour time period?

(a) Show that \( \cos (x^2) \ge \cos x \) for \( 0 \le x \le 1 \). (b) Deduce that \( \displaystyle \int^{\pi/6}_0 \cos (x^2) \, dx \ge \frac{1}{2} \).

An oil storage tank ruptures at time \( t = 0 \) and oil leaks from the tank at a rate of \( r(t) = 100e^{-0.01t} \) liters per minute. How much oil leaks out during the first hour?

Show that $$ 0 \le \int^{10}_5 \frac{x^2}{x^4 + x^2 + 1} \, dx \le 0.1 $$ by comparing the integrand to a simpler function.

A bacteria population starts with 400 bacteria and grows at a rate of \( r(t) = (450.268)e^{1.12567t} \) bacteria per hour. How many bacteria will there be after three hours?

Let \( f(x) = \left\\{ \begin{array}{ll} 0 & \mbox{if \) x < 0 \(}\\\ x & \mbox{if \) 0 \le x \le 1 \(}\\\ 2 - x & \mbox{if \) 1 < x \le 2 \(}\\\ 0 & \mbox{if \) x > 2 \(} \end{array} \right.\) and $$ g(x) = \int^x_0 f(t) \, dt $$ (a) Find an expression for \( g(x) \) similar to the one for \( f(x) \). (b) Sketch the graphs of \( f \) and \( g \). (c) Where is \( f \) differentiable? Where is \( g \) differentiable?

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function \( f(t) = \frac{1}{2} \sin (2\pi t/5) \) has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time \( t \).

Find a function \( f \) and a number \( a \) such that \( \displaystyle 6 + \int^x_a \frac{f(t)}{t^2} \, dt = 2 \sqrt{x} \) for all \( x > 0 \)

Dialysis treatment removes urea and other waste products from a patient's blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which the urea is removed from the blood (in mg/min) is often well described by the equation $$ u(t) = \frac{r}{V} C_0 e^{-rt/V} $$ where \( r \) is the rate of flow of blood through the dialyzer (in mL/min), \( V \) is the volume of the patient's blood (in mL), and \( C_0 \) is the amount of urea in the blood (in mg) at time \( t = 0 \). Evaluate the integral \( \displaystyle \int^{30}_0 u(t) \, dt \) and interpret it.

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after \( t \) weeks is $$ \frac{dx}{dt} = 5000 \biggl( 1 - \frac{100}{(t + 10)^2} \biggr) \text{calculators/week} $$ (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.

A high-tech company purchases a new computing system whose initial value is \( V \). The system will depreciate at the rate \( f = f(t) \) and will accumulate maintenance costs at the rate \( g = g(t) \), where \( t \) is the time measured in months. The company wants to determine the optimal time to replace the system. (a) Let $$ C(t) = \frac{1}{t} \int^t_0 [f(s) + g(s)] \, ds $$ Show that the critical numbers of \( C \) occur at the numbers \( t \) where \( C(t) = f(t) + g(t) \). (b) Suppose that \( f(t) = \left\\{ \begin{array}{ll} \frac{V}{15} - \frac{V}{450}t & \mbox{if \) 0 < t \le 30 \(}\\\ 0 & \mbox{if \) t > 30 \(} \end{array} \right.\) and \( g(t) = \frac{Vt^2}{12,900} \) \( t > 0 \) Determine the length of time \( T \) for the total depreciation \( \displaystyle D(t) = \int^t_0 f(s) \, ds \) to equal the initial value \( V \). (c) Determine the absolute minimum of \( C \) on \( (0, T] \). (d) Sketch the graphs of \( C \) and \( f + g \) in the same coordinate system, and verify the result in part (a) in this case.

If \( f \) is continuous and \( \displaystyle \int^9_0 f(x) \, dx = 4 \), find \( \displaystyle \int^3_0 xf(x^2) \, dx \).

If \( a \) and \( b \) are positive numbers, show that $$ \int^1_0 x^a(1 - x)^b \,dx = \int^1_0 x^b(1 - x)^a \,dx $$

If \( f \) is continuous on \( [0, \pi] \), use the substitution \( u = \pi - x \) to show that $$ \int^{\pi}_0 x f(\sin x) \,dx = \frac{\pi}{2} \int^{\pi}_0 f(\sin x) \,dx $$

(a) If \( f \) is continuous, prove that $$ \int^{\pi/2}_0 f(\cos x) \,dx = \int^{\pi/2}_0 f(\sin x) \,dx $$ (b) Use part (a) to evaluate \( \displaystyle \int^{\pi/2}_0 \cos^2 x \,dx \) and \( \displaystyle \int^{\pi/2}_0 \sin^2 x \,dx \)