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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 4.4 Indefinite Integrals & the Net Change Theorem

Subsection 4.4.1 Before Class

https://mymedia.ou.edu/media/4.4-1/1_zny56xxk
Figure 42. Pre-Class Video 1

Subsubsection 4.4.1.1 Indefinite Integrals

Definition 4.4.1. Indefinite Integral.
The indefinite integral is a family of functions \(F(x)\) such that \(F'(x) = f(x)\) or \(\ds \int f(x)\, dx = F(x)\)
Example 4.4.2.
Write the indefinite integral for \(\ds \int x^2\, dx\)
Solution.
\(\ds \int x^2\, dx = \dfrac{1}{3}x^3 + C\)
Question 4.4.3.
Give an explicit distinction between the definite integral and the indefinite integral.
Solution.
The definite integral always returns a number while the indefinite integral always returns a function.
Useful Indefinite Integrals.
Function Indefinite Integral Function Indefinite Integral
\(\ds \int c\cdot f(x)\, dx\) \(c\cdot F(x) +C\) \(\ds \int [f(x)\pm g(x)]\, dx\) \(F(x) + G(x) + C\)
\(\ds \int k\, dx\) \(kx + C\) \(\ds \int x^n\, dx\) \(\dfrac{x^{n+1}}{n+1} + C\)
\(\ds \sin x\, dx\) \(-\cos x + C\) \(\ds \int \cos x\, dx\) \(\sin x + C\)
\(\ds \int \sec^2 x\, dx\) \(\tan x + C\) \(\ds \int \csc^2 x\, dx\) \(-\cot x + C\)
\(\ds \int \sec x \tan x\, dx\) \(\sec x + C\) \(\ds \int \csc x\cot x\, dx\) \(-\csc x + C\)
Example 4.4.4.
Find the general indefinite integral for \(f(x) = 3x^5-2\csc^2 x\)
Solution.
\(\ds \int f(x)\, dx = \dfrac{1}{2}x^6 + 2\cot x + C\)
Example 4.4.5.
Evaluate \(\ds \int \dfrac{\sin\theta}{\cos^2\theta}\,d\theta\)
Solution.
\(\ds \int \dfrac{\sin\theta}{\cos^2\theta}\,d\theta = \sec\theta + C\)
Example 4.4.6.
Evaluate \(\ds \int (6-2\cos x)\, dx\)
Solution.
\(\ds \int (6-2\cos x)\, dx = 6x-2\sin x + C\)

Subsection 4.4.2 Pre-Class Activities

Example 4.4.7.

Imagine that you are able to give your future self some advice, while you’re studying. Looking back over the notes, how would you describe the difference between a definite integral and an indefinite integral to your future self?
Solution.
Answers vary

Example 4.4.8.

Compute the following:
  1. \(\displaystyle \ds \int \lrpar{\cos x + \dfrac{1}{3}x}\, dx\)
  2. \(\displaystyle \ds \int \lrpar{1-x^2}^2\, dx\)
  3. \(\displaystyle \ds \int_1^2 \lrpar{4x^3 - 3x^2 + 2x}\, dx\)
Solution.
  1. \(\displaystyle \sin x + \dfrac{1}{6}x^2 + C\)
  2. \(\displaystyle x-\dfrac{2}{3}x^3 +\dfrac{1}{5}x^5 + C\)
  3. \(\displaystyle 11\)

Subsection 4.4.3 In Class

Subsubsection 4.4.3.1 The Net Change Theorem

Question 4.4.9.
  1. In Section 4.1, how did we find the accumulated change of a function? Give an real-world example of how those techniques would be used.
  2. In Section 4.3, we learned the Fundamental Theorem of Calculus. Rewrite FTC 2 here.
Solution.
  1. Answers vary
  2. If \(f(x)\) is continuous on \([a,b]\) and \(F(x)\) is an antiderivative of \(f(x)\) on that interval, then \(\ds \int_a^b f(x)\, dx = F(b) - F(a)\)
Question 4.4.11.
What relationship(s) do you see between the Net Change Theorem and FTC 2?
Solution.
We get the same result, but we are using different functions.
Displacement/Distance.
When talking about physical situations, the displacement of a particle is the net change of the particle’s position, and the distance is the total change of the particle’s position.
Example 4.4.12.
A particle moves along a line so that its velocity at time \(t\) is \(v(t) = t^2-t-6\) m/s.
  1. Find the displacement of the particle during the time period \(1\leq t\leq 4\)
  2. Find the distance traveled during this time period.
Solution.
  1. \(-\dfrac{9}{2}\) m
  2. \(\dfrac{61}{6}\) m
Example 4.4.13.
A particle moving along a line has velocity \(v(t) = t^2 -2t-3\) m/s. Find the displacement and the total distance traveled by the particle between 1 and 4 seconds.
Solution.
The displacement is \(-3\) m and the distance traveled is \(\dfrac{23}{3}\) m

Subsubsection 4.4.3.2 Practice

Example 4.4.14.
Find the general indefinite integral of \(f(x) = x^{1.3}-7x^{2.5}\)
Solution.
\(\ds \int f(x)\, dx = \dfrac{x^{2.3}}{2.3} - 2x^{3.5} + C\)
Example 4.4.15.
Find the general indefinite integral of \(f(x) = \sqrt[5]{x^4}\)
Solution.
\(\ds \int f(x)\, dx = \dfrac{5}{9} x^{9/5} + C\)
Example 4.4.16.
Find the general indefinite integral of \(f(x) = \dfrac{1-\sqrt{x}+x}{\sqrt{x}}\)
Solution.
\(\ds \int f(x)\, dx = 2x^{1/2} + \dfrac{2}{3}x^{3/2} + C\)
Example 4.4.17.
Find the general indefinite integral of \(f(x) = 2+\tan^2x\)
Solution.
\(\ds \int f(x)\, dx = x + \tan x + C\)
Example 4.4.18.
Find the general indefinite integral of \(f(t) = \dfrac{1-\sin^3t}{\sin^2t}\)
Solution.
\(\ds \int f(t)\, dt = -\cot t + \cos t + C\)
Example 4.4.19.
Evaluate the integral \(\ds \int_{-2}^3 (x^2-3)\, dx\)
Solution.
\(\ds \int_{-2}^3 (x^2-3)\, dx = -\dfrac{10}{3}\)
Example 4.4.20.
Evaluate the integral \(\ds \int_0^3 (1 + 6w^2-10w^4)\,dw\)
Solution.
\(\ds \int_0^3 (1 + 6w^2-10w^4)\,dw = -429\)
Example 4.4.21.
Evaluate the integral \(\ds \int_0^\pi (4\sin \theta -3\cos\theta)\, d\theta\)
Solution.
\(\ds \int_0^\pi (4\sin \theta -3\cos\theta)\, d\theta =8\)
Example 4.4.22.
Evaluate the integral \(\ds \int_0^{\pi/3} \dfrac{\sin\theta + \sin\theta\tan^2\theta}{\sec^2\theta}\,d\theta\)
Solution.
\(\ds \int_0^{\pi/3} \dfrac{\sin\theta + \sin\theta\tan^2\theta}{\sec^2\theta}\,d\theta = \dfrac{1}{2}\)
Example 4.4.23.
Evaluate the integral \(\ds \int_1^8 \dfrac{2+t}{\sqrt[3]{t^2}}\)
Solution.
\(\ds \int_1^8 \dfrac{2+t}{\sqrt[3]{t^2}} = \dfrac{69}{4}\)
Example 4.4.24.
A honeybee population starts with 100 bees and increases at a rate of \(n'(t)\) bees per week. What does 100 + \(\ds \int_0^{15}n'(t)\,dt/\) represent?
Solution.
It gives the total number of bees after 15 weeks.
Example 4.4.25.
If \(x\) is measured in meters and \(f(x)\) is measured in newtons, what are the units of \(\ds \int_0^{100}f(x)\,dx\text{?}\)
Solution.
Newton\(\cdot\)meters
Example 4.4.26.
The acceleration function of a particle is \(a(t) = t+4\) m/s\(^2\text{,}\) and its the initial velocity is 5 m/s. Find the velocity at time \(t\text{,}\) and the distance traveled between time \(t = 0\) and \(t = 5\text{.}\)
Solution.
The velocity is \(v(t) = \dfrac{1}{2}t^2 + 4t + 8\) and the particle travels \(\dfrac{665}{6}\) m.

Subsection 4.4.4 After Class Activities

Example 4.4.27.

Let \(r(\theta) = \dfrac{1+\cos^2\theta}{\cos^2\theta}\text{.}\) Find the indefinite integral \(\ds \int r(\theta)\, d\theta\) and the definite integral on the interval \([0,\pi/4]\text{.}\)
Solution.
\(\ds \int r(\theta)\, d\theta = \tan \theta + \theta + C\) and \(\ds \int_0^{\pi/4} r(\theta)\, d\theta = 1 + \dfrac{\pi}{4}\)

Example 4.4.28.

If \(f(x)\) is the slope of a trail at a distance of \(x\) miles from the start of the trail, what does \(\ds \int_3^5 f(x)\, dx\) represent?
Solution.
The net change in height between 3 and 5 miles from the start of the trail.

Example 4.4.29.

The current in a wire is defined to be the derivative of charge, i.e. \(I(t) = Q'(t)\text{.}\) What does \(\ds \int_a^b I(t)\, dt\) represent?
Solution.
The net change in charge between \(t=a\) and \(t=b\)

Example 4.4.30.

A particle moving along a line has acceleration given by \(a(t) = 2t+3\text{.}\) If \(v(0) = -4\text{,}\) find the particle’s velocity and distance traveled in the first three seconds of motion.
Solution.
\(v(t) = t^2 + 3t-4\) and the distance traveled is \(\dfrac{89}{6}\text{.}\)
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