More Results about Definite Integrals

Section 5.5 More Results about Definite Integrals

We’ll end this chapter by looking a bit more closely at definite integrals and pulling a couple of small results out of our understanding of them, as well as some prior knowledge.

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Subsection Symmetry

Activity 5.5.1. Symmetry in Functions and Integrals.

First, let’s take a moment to remind ourselves (or see for the first time) what two types of “symmetry” we’ll be considering. We call them “even” and “odd” symmetry, but sometimes we think of them as a “reflective” symmetry and a “rotational” symmetry in the graphs of our functions.

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(a)

Convince yourself that you know what we mean when we say that a function is even symmetric on an interval if \(f(-x)=f(x)\) on the interval.

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Similarly, convince yourself that you know what we mean when we say that a function is odd symmetric on an interval if \(f(-x)=-f(x)\) on the interval.

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Hint.

Look at the relationship between the points on the graphs when you select the different symmetries: How do their \(x\)-values relate to each other? How do their \(y\)-values relate to each other?

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(b)

Now let’s think about areas. Before we visualize too much, let’s start with a small question: How does the height of a function impact the area defined by a definite integral? It should be helpful to think about Riemann sums and areas of rectangles here.

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The important question then, is how does a function being even or odd symmetric tell us information about areas defined by definite integrals of that function?

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Hint.

If we know that for an even symmetric function, there are some heights/\(y\)-values that are the same, then we know that there are some areas/integrals that should also be the same. Which ones?

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If we know that for an odd symmetric function, there are some heights/\(y\)-values that are opposite, then we know that there are some areas/integrals that should also be opposite. Which ones?

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Theorem 5.5.1. Definite Integrals of Symmetric Functions.

If \(f(x)\) is a continuous function on \([-a,a]\) for some real number \(a\gt 0\text{,}\) then:

If \(f(x)\) is even symmetric on \([-a,a]\text{,}\) then:

\begin{equation*} \int_{x=-a}^{x=0} f(x)\;dx = \int_{x=0}^{x=a}f(x)\;dx\text{.} \end{equation*}

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If \(f(x)\) is odd symmetric on \([-a,a]\text{,}\) then:

\begin{equation*} \int_{x=-a}^{x=0} f(x)\;dx = -\int_{x=0}^{x=a}f(x)\;dx\text{.} \end{equation*}

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Activity 5.5.2. Connecting Symmetric Integrals.

We’re going to do some sketching here, and I want you to be clear about something: your sketches can be absolutely terrible. It’s ok! They just need to embody the kind of symmetry we’re talking about. You will probably sketch something and notice that your areas aren’t to scale (or maybe even the wrong sign!), and that’s fine.

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It might be helpful to practice sketching graphs accurately, but don’t worry if that part is a struggle.

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(a)

Sketch a function \(f(x)\) with the following properties:

\(f(x)\) is even symmetric on the interval \([-6,6]\)
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\(\displaystyle \displaystyle\int_{x=0}^{x=6} f(x)\;dx = 4\)
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\(\displaystyle \displaystyle \int_{x=-6}^{x=-2}f(x)\;dx = -1\)
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(b)

Find the values of the following integrals:

\(\displaystyle \displaystyle\int_{x=0}^{x=2} f(x)\;dx\)
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\(\displaystyle \displaystyle\int_{x=-6}^{x=6} f(x)\;dx\)
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Hint.

Since \(f(x)\) is even symmetric, what are the two other integrals that we know about? How can we use those to help us find these two?

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Answer.

\(\displaystyle \displaystyle\int_{x=0}^{x=2} f(x)\;dx = 5\)
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\(\displaystyle \displaystyle\int_{x=-6}^{x=6} f(x)\;dx = 8\)
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(c)

Sketch a function \(g(x)\) with the following properties:

\(g(x)\) is odd symmetric on the interval \([-9,9]\)
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\(\displaystyle \displaystyle\int_{x=0}^{x=4} g(x)\;dx = 5\)
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\(\displaystyle \displaystyle \int_{x=-9}^{x=0}g(x)\;dx = 2\)
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(d)

Find the values of the following integrals:

\(\displaystyle \displaystyle\int_{x=-9}^{x=-4} g(x)\;dx\)
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\(\displaystyle \displaystyle\int_{x=-4}^{x=9} g(x)\;dx\)
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Hint.

Since \(g(x)\) is even symmetric, what are the two other integrals that we know about? How can we use those to help us find these two?

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Answer.

\(\displaystyle \displaystyle\int_{x=-9}^{x=-4} g(x)\;dx = 7\)
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\(\displaystyle \displaystyle\int_{x=-4}^{x=9} g(x)\;dx = -2\)
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Subsection Average Value of a Function

Activity 5.5.3. Visualizing the Average Height of a Function.

We are going to build a formula to find the“average height” or “average value” of a function \(f(x)\) on the interval \([a,b]\text{.}\) We’re going to look at a function and try to find the average height. Along the way, we’ll think a bit about areas!

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(a)

Consider the following function. Find the average height of the function on the interval pictured!

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(b)

How does the area “under” the curve \(f(x)\) on the interval compare to the area of the rectangle formed by the average height line?

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(c)

How do you define the two areas?

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Hint.

One of these is the area under \(f(x)\) from \(x=a\) to \(x=b\text{,}\) which we can use calculus for!

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The other is the area of a rectangle with a height (the average height of \(f(x)\)) and a width (the width of the interval).

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(d)

Set up an equation connecting the two areas, and solve for the average height of \(f(x)\text{.}\)

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Hint.

If \(\text{area} = \text{height}\times\text{width}\text{,}\) then doesn’t it make sense that \(\text{height} = \dfrac{\text{area}}{\text{width}}\text{?}\) How, then, do we find average height by dividing an area and a width?

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Theorem 5.5.2. Average Value of a Function.

If a function \(f(x)\) is continuous on the interval \([a,b]\text{,}\) then the average value of \(f(x)\) on \([a,b]\) is:

\begin{equation*} \frac{\displaystyle\int_{x=a}^{x=b}f(x)\;dx}{b-a}\text{.} \end{equation*}

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Example 5.5.3.

A small model glider airplane is thrown and travels for 10 seconds before it hits the ground. The height of the glider is modeled by the function \(h(t)=6+\dfrac{7t}{5}-\dfrac{t^2}{5}\) on the interval \([0,10]\text{.}\)

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Find the average height of the glider on the time interval.

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Solution.

\begin{align*} \frac{1}{10}\int_{t=0}^{t=10} 6+\dfrac{7t}{5}-\dfrac{t^2}{5}\;dt \amp = \frac{1}{10}\left(6t+\frac{7t^2}{10}-\frac{t^3}{15}\right)\bigg|_{t=0}^{t=10} \\ \amp = \frac{1}{10}\left(\frac{190}{3}\right)\\ \amp = \frac{190}{30} \end{align*}

The glider has an average height of 6 feet and 4 inches.

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Practice Problems Practice Problems

1.

Explain how we can use the different types of symmetry can help us as we evaluate the definite integral \(\displaystyle\int_{x=-a}^{x=a} f(x)\;dx\text{.}\)

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2.

Use the definition of even symmetric (where \(f(-x) = f(x)\) for all \(x\)-values) to explain why \(\displaystyle\int_{x=-4}^{x=4} f(x)\;dx = 2\displaystyle\int_{x=0}^{x=4}f(x)\;dx\) for an even symmetric function \(f(x)\text{.}\)

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3.

Use the definition of odd symmetric (where \(f(-x) = -f(x)\) for all \(x\)-values) to explain why \(\displaystyle\int_{x=-4}^{x=4} f(x)\;dx = 0\) for an odd symmetric function \(f(x)\text{.}\)

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4.

Assume that \(f(x)\) is some even symmetric function with \(\displaystyle\int_{x=0}^{x=6} f(x)\;dx = 4\) and \(\displaystyle\int_{x=-3}^{x=0} f(x) \;dx= 9\text{.}\)

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(a)

Sketch a possible graph of the function \(f(x)\text{,}\) and shade in the areas from the integrals listed.

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(b)

Evaluate \(\displaystyle\int_{x=0}^{x=3} f(x)\;dx\text{.}\)

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(c)

Evaluate \(\displaystyle\int_{x=3}^{x=6} f(x)\;dx\text{.}\)

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(d)

Evaluate \(\displaystyle\int_{x=-6}^{x=-3} f(x)\;dx\text{.}\)

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(e)

Evaluate \(\displaystyle\int_{x=-6}^{x=6} f(x)\;dx\text{.}\)

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5.

Assume that \(g(x)\) is some odd symmetric function with \(\displaystyle\int_{x=0}^{x=6} g(x)\;dx = 4\) and \(\displaystyle\int_{x=-3}^{x=0} g(x) \;dx= 9\text{.}\)

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(a)

Sketch a possible graph of the function \(g(x)\text{,}\) and shade in the areas from the integrals listed.

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(b)

Evaluate \(\displaystyle\int_{x=0}^{x=3} g(x)\;dx\text{.}\)

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(c)

Evaluate \(\displaystyle\int_{x=3}^{x=6} g(x)\;dx\text{.}\)

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(d)

Evaluate \(\displaystyle\int_{x=-6}^{x=-3} g(x)\;dx\text{.}\)

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(e)

Evaluate \(\displaystyle\int_{x=-6}^{x=6} g(x)\;dx\text{.}\)

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6.

Explain how to find the average height of a function on an interval. Why does this work?

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7.

Note that \(y=\sin(x)\) is odd symmetric and \(y=x^2\) is even symmetric.

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(a)

Show that \(f(x) = x^2\sin(x)\) is odd symmetric by showing that \(f(-x) = -f(x)\text{.}\)

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(b)

Evaluate \(\displaystyle\int_{x=-\pi}^{x=\pi} x^2\sin(x)\;dx\text{.}\)

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Hint.

You shouldn’t need to find an antiderivative for this function, which is good news (since this is a tricky function to antidifferentiate).

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(c)

Find the average height of \(f(x)\) on the interval \([-\pi, \pi]\text{.}\)

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8.

Note that \(y=\cos(x)\) is even symmetric and \(y=x^2\) is even symmetric.

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(a)

Show that \(f(x) = x^2\cos(x)\) is even symmetric by showing that \(f(-x) = f(x)\text{.}\)

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(b)

Given that \(\displaystyle\int_{x=0}^{x=\pi} x^2\cos(x)\;dx = -2\pi\text{,}\) evaluate \(\displaystyle\int_{x=-\pi}^{x=\pi} x^2\cos(x)\;dx\text{.}\)

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(c)

Find the average height of \(f(x)\) on the interval \([-\pi,\pi]\text{.}\)

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9.

Find the average height of the following functions on the intervals listed.

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(a)

\(f(x) = e^x -x\) on \([0,3]\)

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(b)

\(g(x) = \cos(x) + x\) on \(\left[0,\frac{\pi}{2}\right]\)

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(c)

\(j(x) = \dfrac{1}{1+x^2}\) on \([-1,1]\)

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(d)

\(\ell(x) = \dfrac{1}{x^2}\) on \([1,4]\)

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