Let \(f(t)\) be a continuous function defined on \([a,b]\text{.}\) The definite integral \(\int_a^b f(x)\ dx\) is the “area under \(f\)” on \([a,b]\text{.}\) We can turn this concept into a function by letting the upper (or lower) bound vary.
Let \(F(x) = \int_a^x f(t)\ dt\text{.}\) It computes the area under \(f\) on \([a,x]\) as illustrated in Figure 5.4.1. We can study this function using our knowledge of the definite integral. For instance, \(F(a)=0\) since \(\int_a^af(t)\ dt=0\text{.}\)
Consider \(f(t)=2t\) pictured in Figure Figure 5.4.3 and its associated “area so far” function, \(F(x)=\int_1^x 2t\ dt\text{.}\) Using the graph of \(f\) and geometry, find an explicit formula for \(F\text{.}\)
As Example 5.4.2 hinted, we can apply calculus ideas to \(F(x)\text{;}\) in particular, we can compute its derivative. In Example 5.4.2, \(F(x)=x^2-1\text{,}\) so \(F'(x)=2x=f(x)\text{.}\) While this may seem like an innocuous thing to do, it has far–reaching implications, as demonstrated by the fact that the result is given as an important theorem.
Let \(f\) be continuous on \([a,b]\) and let \(F(x) = \int_a^x f(t)\ dt\text{.}\) Then \(F\) is a differentiable function on \((a,b)\text{,}\) and \begin{equation*} \Fp(x)=f(x). \end{equation*} In other words: \begin{equation*} \lzoo{x}{\int_a^x f(t)\ dt}=f(x). \end{equation*}
Initially this seems simple, as demonstrated in the following example.
Let \(\ds F(x) = \int_{-5}^x (t^2+\sin(t) )\ dt\text{.}\) What is \(\Fp(x)\text{?}\)
What we have done in Example 5.4.7 was more than finding a complicated way of computing an antiderivative. Consider a function \(f\) defined on an open interval containing \(a\text{,}\) \(b\) and \(c\text{.}\) Suppose we want to compute \(\int_a^b f(t)\ dt\text{.}\) First, let \begin{equation} F(x) = \int_c^x f(t)\ dt\text{.}\label{eq_ftc_c}\tag{5.4.1} \end{equation} Using the properties of the definite integral found in Theorem 5.2.9, we know \begin{align*} \int_a^b f(t)\ dt \amp = \int_a^c f(t)\ dt + \int_c^b f(t)\ dt\\ \amp = -\int_c^a f(t)\ dt + \int_c^b f(t)\ dt \end{align*} Using Equation (5.4.1), let \(x=a\) in the first integral and \(x=b\) in the second integral so that \(\int_c^a f(t)\ dt =F(a)\) and \(\int_c^b f(t)\ dt =F(b)\text{.}\) Therefore: \begin{align*} \int_a^b f(t)\ dt \amp =-F(a) + F(b)\\ \amp = F(b) - F(a). \end{align*}
We now see how indefinite integrals and definite integrals are related: we can evaluate a definite integral using antiderivatives! In fact, this is exactly what we noticed in Example 5.4.2. The “area so far” function was indeed an anti-derivative of the integrand. This is the second part of the Fundamental Theorem of Calculus.
Let \(f\) be continuous on \([a,b]\) and let \(F\) be any antiderivative of \(f\text{.}\) Then \begin{equation*} \int_a^b f(x)\ dx = F(b) - F(a). \end{equation*}
We spent a great deal of time in the previous section studying \(\int_0^4(4x-x^2)\ dx\text{.}\) Using the Fundamental Theorem of Calculus, evaluate this definite integral.
Notation: A special notation is often used in the process of evaluating definite integrals using the Fundamental Theorem of Calculus. Instead of explicitly writing \(F(b)-F(a)\text{,}\) the notation \(F(x)\Big|_a^b\) is used. Thus the solution to Example 5.4.9 would be written as: \begin{align*} \int_0^4(4x-x^2)\ dx \amp = \left.\left(2x^2-\frac13x^3\right)\right|_0^4 \\ \amp = \big(2(4)^2-\frac134^3\big)-\big(0-0\big) = 32/3. \end{align*}
The Constant \(C\text{:}\) Any antiderivative \(F(x)\) can be chosen when using the Fundamental Theorem of Calculus to evaluate a definite integral, meaning any value of \(C\) can be picked. The constant always cancels out of the expression when evaluating \(F(b)-F(a)\text{,}\) so it does not matter what value is picked. This being the case, we might as well let \(C=0\text{.}\)
Evaluate the following definite integrals.
We established, starting with Key Idea 2.2.3, that the derivative of a position function is a velocity function, and the derivative of a velocity function is an acceleration function. Now consider definite integrals of velocity and acceleration functions. Specifically, if \(v(t)\) is a velocity function, what does \(\ds \int_a^b v(t)\ dt\) mean?
The Fundamental Theorem of Calculus states that \begin{equation*} \int_a^b v(t)\ dt = V(b) - V(a), \end{equation*} where \(V(t)\) is any antiderivative of \(v(t)\text{.}\) Since \(v(t)\) is a velocity function, \(V(t)\) must be a position function, and \(V(b) - V(a)\) measures a change in position, or displacement.
A ball is thrown straight up with velocity given by \(v(t) = -32t+20\)ft/s, where \(t\) is measured in seconds. Find, and interpret, \(\ds \int_0^1 v(t)\ dt\) and \(\int_0^1 \abs{v(t)}\ dt\text{.}\)
Integrating a rate of change function gives total change. Velocity is the rate of position change; integrating velocity gives the total change of position, i.e., displacement.
Integrating a speed function gives a similar, though different, result. Speed is also the rate of position change, but does not account for direction. That is, the speed an object is the absolute value of its velocity. This is what we saw in Example 5.4.11 when we evaluated \(\int_0^1 \abs{v(t)}\ dt\text{.}\) So integrating a speed function gives total change of position, without the possibility of “negative position change.” Hence the integral of a speed function gives distance traveled.
As acceleration is the rate of velocity change, integrating an acceleration function gives total change in velocity. We do not have a simple term for this analogous to displacement. If \(a(t) = 5\)miles/h\(^2\) and \(t\) is measured in hours, then \begin{equation*} \int_0^3 a(t)\ dt = 15 \end{equation*} means the velocity has increased by 15m/h from \(t=0\) to \(t=3\text{.}\)
Part 1 of the Fundamental Theorem of Calculus (FTC) states that given \(\ds F(x) = \int_a^x f(t)\ dt\text{,}\) we have \(\Fp(x) = f(x)\text{.}\) Using other notation, \(\ds \lzoo{x}{F(x)}=\lzoo{x}{\int_a^x f(t)\ dt} = f(x)\text{.}\) While we have just practiced evaluating definite integrals, sometimes finding antiderivatives is impossible and we need to rely on other techniques to approximate the value of a definite integral. Functions written as \(F(x) = \int_a^x f(t)\ dt\) are useful in such situations.
It may be of further use to compose such a function with another. As an example, we may compose \(F(x)\) with \(g(x)\) to get \begin{equation*} F\big(g(x)\big) = \int_a^{g(x)} f(t)\ dt. \end{equation*}
What is the derivative of such a function? The Chain Rule can be employed to state \begin{equation*} \frac{d}{dx}\Big(F\big(g(x)\big)\Big) = \Fp\big(g(x)\big)g'(x) = f\big(g(x)\big)g'(x). \end{equation*}
An example will help us understand this.
Find the derivative of \(\ds F(x) = \int_2^{x^2} \ln(t) \ dt\text{.}\)
Practice this once more.
Find the derivative of \(\ds F(x) = \int_{\cos(x) }^5 t^3\ dt.\)
Consider continuous functions \(f(x)\) and \(g(x)\) defined on \([a,b]\text{,}\) where \(f(x) \geq g(x)\) for all \(x\) in \([a,b]\text{,}\) as demonstrated in Figure 5.4.16. What is the area of the shaded region bounded by the two curves over \([a,b]\text{?}\)
The area can be found by recognizing that this area is “the area under \(f\) \(-\) the area under \(g\text{.}\)” Using mathematical notation, the area is \begin{equation*} \int_a^b f(x)\ dx - \int_a^b g(x)\ dx. \end{equation*}
Properties of the definite integral allow us to simplify this expression to \begin{equation*} \int_a^b\big(f(x) - g(x)\big)\ dx. \end{equation*}
Let \(f(x)\) and \(g(x)\) be continuous functions defined on \([a,b]\) where \(f(x)\geq g(x)\) for all \(x\) in \([a,b]\text{.}\) The area of the region bounded by the curves \(y=f(x)\text{,}\) \(y=g(x)\) and the lines \(x=a\) and \(x=b\) is \begin{equation*} \int_a^b \big(f(x)-g(x)\big)\ dx. \end{equation*}
Find the area of the region enclosed by \(y=x^2+x-5\) and \(y=3x-2\text{.}\)
Consider the graph of a function \(f\) in Figure 5.4.20 and the area defined by \(\int_1^4 f(x)\ dx\text{.}\) Three rectangles are drawn in Figure 5.4.21; in Figure 5.4.21.(a), the height of the rectangle is greater than \(f\) on \([1,4]\text{,}\) hence the area of this rectangle is is greater than \(\int_0^4 f(x)\ dx\text{.}\)
In Figure 5.4.21.(b), the height of the rectangle is smaller than \(f\) on \([1,4]\text{,}\) hence the area of this rectangle is less than \(\int_1^4 f(x)\ dx\text{.}\)
Finally, in Figure 5.4.21.(c) the height of the rectangle is such that the area of the rectangle is exactly that of \(\int_0^4 f(x)\ dx\text{.}\) Since rectangles that are “too big”, as in (a), and rectangles that are “too little,” as in (b), give areas greater/lesser than \(\int_1^4 f(x)\ dx\text{,}\) it makes sense that there is a rectangle, whose top intersects \(f(x)\) somewhere on \([1,4]\text{,}\) whose area is exactly that of the definite integral.
We state this idea formally in a theorem.
Let \(f\) be continuous on \([a,b]\text{.}\) There exists a value \(c\) in \([a,b]\) such that \begin{equation*} \int_a^bf(x)\ dx = f(c)(b-a). \end{equation*}
This is an existential statement; \(c\) exists, but we do not provide a method of finding it. Theorem 5.4.22 is directly connected to the Mean Value Theorem of Differentiation, given as Theorem 3.2.3; we leave it to the reader to see how.
The Theorem 5.4.22 simply says that there is a rectangle with height \(f(c)\) and width \(b-a\text{,}\) the area of which is the same as the area between \(f\) and the \(x\)-axis over\([a,b]\text{.}\) Furthermore, we know that \(c\) will be in the interval \([a,b]\text{.}\)
We demonstrate the principles involved in this version of the Mean Value Theorem in the following example.
Consider \(\int_0^\pi \sin(x) \ dx\text{.}\) Find a value \(c\) guaranteed by the Mean Value Theorem.
We now turn our attention to a related topic —average value. Let \(f\) be a function on \([a,b]\) with \(c\) such that \(f(c)(b-a) = \int_a^bf(x)\ dx\text{.}\) Consider \(\int_a^b\big(f(x)-f(c)\big)\ dx\text{:}\) \begin{align*} \int_a^b\big(f(x)-f(c)\big)\ dx \amp = \int_a^b f(x) - \int_a^b f(c)\ dx\\ \amp = f(c)(b-a) - f(c)(b-a)\\ \amp = 0. \end{align*}
When \(f(x)\) is shifted by \(-f(c)\text{,}\) the amount of area under \(f\) above the \(x\)–axis on \([a,b]\) is the same as the amount of area below the \(x\)–axis above \(f\text{;}\) see Figure 5.4.25 for an illustration of this. In this sense, we can say that \(f(c)\) is the average value of \(f\) on \([a,b]\text{.}\)
The value \(f(c)\) is the average value in another sense. First, recognize that the Mean Value Theorem can be rewritten as \begin{equation*} f(c) = \frac{1}{b-a}\int_a^b f(x)\ dx, \end{equation*} for some value of \(c\) in \([a,b]\text{.}\) Replacing the integral with the limit of a Riemann sum (as in Theorem 5.3.20): \begin{align*} f(c) \amp = \frac{1}{b-a}\int_a^b f(x)\ dx \amp \\ \amp = \frac{1}{b-a} \lim_\limits{n \to \infty}\sum_{i=1}^n f(c_i)\ \Delta x \amp \text{Using } \knowl{./knowl/thm_riemann_sum.html}{\text{Theorem 5.3.20}} \\ \amp =\frac{1}{b-a} \lim_\limits{n \to \infty}\sum_{i=1}^n f(c_i)\ \frac{b-a}{n}\amp \Delta x =\frac{b-a}{n}\\ \amp =\lim_\limits{n \to \infty}\sum_{i=1}^n f(c_i)\frac{1}{n} \amp \text{Cancelling the common factor of } b-a. \end{align*}
Examining this last line closely, the expression \(\sum_{i=1}^n f(c_i)\frac{1}{n}\) represents adding up \(n\) sample values of \(f(x)\)and then dividing by \(n\text{.}\) This is exactly what we do when we calculate the average of a set of \(n\) numbers. Now when we consider taking the limit as \(n\) goes to \(\infty\text{,}\) \(\lim_\limits{n \to \infty}\sum_{i=1}^n f(c_i)\frac{1}{n}\text{,}\) we are adding up all of the function's output values over \([a,b]\) and dividing by the “number of numbers”. In a sense, we are adding up an infinite number of output values and then dividing by the number of terms we summed (which is again infinite).
This leads us to a definition.
Let \(f\) be continuous on \([a,b]\text{.}\) The average value of \(f\) on \([a,b]\) is \(f(c)\text{,}\) where \(c\) is a value in \([a,b]\) guaranteed by the Mean Value Theorem. I.e., \begin{equation*} \text{ Average Value of \(f\) on \([a,b]\) } = \frac{1}{b-a}\int_a^b f(x)\ dx. \end{equation*}
An application of this definition is given in the following example.
An object moves back and forth along a straight line with a velocity given by \(v(t) = (t-1)^2\) on \([0,3]\text{,}\) where \(t\) is measured in seconds and \(v(t)\) is measured in ft/s.
What is the average velocity of the object?
We can understand the above example through a simpler situation. Suppose you drove 100 miles in 2 hours. What was your average speed? The answer is simple: displacement/time = 100 miles/2 hours = 50 mph.
What was the displacement of the object in Example 5.4.27? We calculate this by integrating its velocity function: \(\int_0^3 (t-1)^2\ dt = 3\) ft. Its final position was 3 feet from its initial position after 3 seconds: its average velocity was 1 ft/s.
This section has laid the groundwork for a lot of great mathematics to follow. The most important lesson is this: definite integrals can be evaluated using antiderivatives. Since Section 5.3 established that definite integrals are the limit of Riemann sums, we can later create Riemann sums to approximate values other than “area under the curve,” convert the sums to definite integrals, then evaluate these using the The Fundamental Theorem of Calculus, Part 2. This will allow us to compute the work done by a variable force, the volume of certain solids, the arc length of curves, and more.
The downside is this: generally speaking, computing antiderivatives is much more difficult than computing derivatives. Chapter 6 is devoted to techniques of finding antiderivatives so that a wide variety of definite integrals can be evaluated. Before that, Section 5.5 explores techniques of approximating the value of definite integrals beyond using the Left Hand, Right Hand and Midpoint Rules.
Terms and Concepts
In the following exercises, evaluate the definite integral.