4.2.3What is the FTC?

The Fundamental Theorem of Calculus

4-65.

Anita is calmly sketching horizontal lines in her notebook when she notices a pattern among the area functions under horizontal lines. For example, the area function under the line $y = 2$ can be found with the definite integral $A ( x ) = \int _ { a } ^ { x } 2 d t$ . Sketch a graph of $y = 2$ and use it to investigate how the value of the lower bound, $a$ , affects the equation of the area function.

If $a = 0$ , write the equation for $A\left(x\right)$ .
If $a = 1$ , write the equation for $A\left(x\right)$ .
If $a = 10$ , write the equation for $A\left(x\right)$ .
If $a = –5$ , write the equation for $A\left(x\right)$ .
Examine the area functions you wrote in parts (a) through (d). How are they the same? How are they different? In particular, how are they all related to the original function $y = 2$ ?

4-66.

Periodic curve, labeled, f of x = cosine x, x axis scaled from negative 1 sixth pi to 1 half pi, max at the point (0, comma 1), with shaded region below the curve, above the x axis, & between x values, 1 sixth pi, & 1 third pi. Anita’s teammate Tommy is stuck. He wants to evaluate $\int _ { \pi / 6 } ^ { \pi / 3 } \operatorname { cos } ( x ) d x$ , but he does not have a calculator.

Tommy knows that this integral can be evaluated with $A ( \frac { \pi } { 3 } ) - A ( \frac { \pi } { 6 } )$ . Unfortunately, he does not know the equation for $A\left(x\right)$ . Thinking about her answer to part (e) of problem 4-65, Anita suggests finding some function whose derivative is equal to $\cos\left(x\right)$ .
Tommy objects. “I can think of more than one function like that!”
What is Tommy talking about? List four functions whose derivative is $\cos\left(x\right)$ . In other words, list four different antiderivatives of $\cos\left(x\right)$ .
“Don’t worry, Tommy,” says Anita in a calm, comforting voice. “You can use any of those antiderivatives!”
Try it. Each member of your team should choose a different antiderivative for $y=\cos\left(x\right)$ and use it to evaluate $\int_{\pi/6}^{\pi/3}\cos(x)dx=$ . $A(\frac{\pi}{3})-A(\frac{\pi}{6})=$ Compare your results.
Tommy and Anita are delighted by their fundamental discovery. They now have a procedure to calculate the exact area under the curve of any function! “From now on,” announces Tommy, “I will not worry about the constant when evaluating a definite integral. I will write my antiderivative with a $+C$ instead.”
Explain why all area functions lead to the same result when evaluating a definite integral.

4-67.

Use Tommy and Anita’s technique to evaluate $\int_1^2(6x^2+7)dx$ . Test your results using a graphing calculator.

4-68.

First quadrant, unscaled x axis labeled, t, with one tick mark, labeled x, 5 sixths of the way right, increasing line labeled, f of t = 2 t + 5, passing through the point (0, comma 5), with shaded region labeled, A of x, below line & left of x axis tick mark. THE FUNDAMENTAL THEOREM OF CALCULUS

Use geometry to write an equation for the area of the shaded region in the graph at right.
That is, what is $\int _ { 0 } ^ { x } ( 2 t + 5 ) d t$ ?
What is $A^\prime\left(x\right)$ ? Compare it to the original function.
Part (b) shows that the derivative of an area function is the original function. Test this idea on a general linear function, $f\left(x\right) = mx + b$ , by determining $A\left(x\right)$ and $A^\prime\left(x\right)$ . Does the same result happen?

4-69.

First quadrant, generic curve labeled, y = f of x, 3 tick marks on x axis, labeled, a, x, & x + h, which is very close to x, light gray shaded region, below the curve, right of, a, & left of, x, & dark gray shaded rectangle, base on x axis between, x & x + h, top left vertex on the curve, left side labeled, f of x. The relationship between derivatives and integrals is fundamental to calculus, and the diagram at right seeks to illustrate that relationship. Through carefully analyzing this diagram, you will discover a way to simplify the derivative of an integral:

$\frac { d } { d x } \int _ { a } ^ { x } f ( t ) d t$ = ?

Start by examining the dark gray rectangle in the diagram. As $h → 0$ , what does that dark gray rectangle represent relative to $\int _ { a } ^ { x } f ( t ) d t$ ?
Write two different expressions that can be used to calculate the area of the dark gray rectangle as $h → 0$ . One expression should use a familiar geometric formula and the other should use an integral. Do not forget that each expression must include the limit as $h → 0$ . Note—neither of these limits should be evaluated, yet.
Since both expressions in part (b) represent the same area, combine them into a single equation.
Knowing that $F$ represents the antiderivative of $f$ , use the method developed in problem 4-66 to evaluate the integral part of your equation.
Algebraically maneuver your equation so that one side represents the derivative of $\int _ { a } ^ { x } f ( t ) d t$ in limit form. Recall that a limit as $h → 0$ is often used to define a derivative. Be prepared to explain your result to the class.
Evaluate the limits.
Using your answer to part (f), what does $\frac { d } { d x } \int _ { a } ^ { x } f ( t ) d t$ equal?

Math Notes

Indefinite Integrals

Unlike definite integrals, which are defined between two bounds, indefinite integrals have no bounds and are written in the form $\int f ( x ) d x$ .

The indefinite integral $\int f ( x ) d x$ represents all of the antiderivatives, $F ( x ) = \int _ { a } ^ { x } f ( t ) d t$ , that are associated with every possible value of $a$ . For that reason, an indefinite integral has infinitely many antiderivatives, each differing by a constant.

$\int f ( x ) d x = F ( x ) + C$

Example: $\int ( 2 x + 3 ) d x = x ^ { 2 } + 3 x + C$ because $\frac { d } { d x } ( x ^ { 2 } + 3 x + C ) = 2 x + 3$ for any $C$ .
Note: Not every function has a closed form antiderivative.

Review and Preview problems below

4-70.

Rewrite the following integral expressions as a single integral. Homework Help ✎

$\int _ { - 5 } ^ { 2 } 6 x ^ { 3 } d x - \int _ { - 5 } ^ { 2 } - 9 x d x$
$\int _ { 5 } ^ { 9 } h ( x ) d x + \int _ { 9 } ^ { 3 } h ( x ) d x$
$\int _ { 2 } ^ { 7 } \pi ( 2 x ) ^ { 2 } d x + \int _ { 2 } ^ { 7 } \pi ( \sqrt { x } ) ^ { 2 } d x$
$\int _ { 1 } ^ { 4 } ( x + 5 ) ^ { 2 } d x + \int _ { 6 } ^ { 9 } \sqrt { x } d x$

4-71.

Keily put his dog on a diet! If $f\left(t\right)$ represents pounds lost per day and $t$ is measured in days, what does $\int _ { a } ^ { b } f ( t ) d t$ represent? What are its units? Homework Help ✎

4-72.

The graph of a function $y = f\left(x\right)$ is shown at the right. Use the graph to evaluate the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow - 1 } f ( x )$
$\lim\limits _ { x \rightarrow 2 } f ( x )$
$\lim\limits _ { x \rightarrow 2 ^ { - } } f ( x )$
$\lim\limits _ { x \rightarrow 2 ^ { + } } f ( x )$
$\lim\limits _ { x \rightarrow 5 } f ( x )$
$\lim\limits _ { x \rightarrow \infty } f ( x )$
Where (if anywhere) does the derivative of $f$ not exist?

4-73.

The graph at right shows the velocity of an object over time defined by the function $v\left(t\right) = −0.5t^{2} + 3t + 1$ . Homework Help ✎

Use your graphing calculator to evaluate $\int _ { 0 } ^ { 4 } ( - 0.5 t ^ { 2 } + 3 t + 1 ) d t$ .
What does the result in part (a) calculate?
What does $\frac { d } { d t } ( v ( t ) )$ represent?
What will the units be in part (c)?

First quadrant, x axis labeled time, seconds, y axis labeled velocity, feet per sec, downward parabola, vertex at (3, comma 5), passing through (0, comma 1), with shaded region below the curve & left of x = 4.

4-74.

If $\int _ { 2 } ^ { 4 } f ( x ) d x = 10$ evaluate: Homework Help ✎

$\int _ { 4 } ^ { 2 } - f ( x ) d x$
$\int _ { 10 } ^ { 12 } ( f ( x - 8 ) + 4 ) d x$
$\int _ { 10 } ^ { 12 } f ( x + 8 ) d x$

4-75.

Given $f\left(x\right) = 2x$ , write the equation of a vertical line that will divide $\int_0^{10}f\left(x\right)dx$ in half. Homework Help ✎

4-76.

Compare how distance and velocity are related with these two scenarios: Homework Help ✎

As an arrow flies through the air, the distance it has traveled in feet at time $t$ is given by $s ( t ) = 4 \sqrt { t }$ . Without your calculator, determine the velocity, $s^\prime\left(t\right)$ , at times $t = 1, 4,$ and $16$ seconds. Explain what concepts of calculus you applied in order to solve this problem.
As a train travels past a station, its velocity, measured in miles per hour, is $v\left(t\right) = 9t + 32$ . If the train is directly across from the station when $t = 0$ , determine the position of the train at $t = 1$ hour. Explain what concepts of calculus you applied in order to solve this problem.
Parts (a) and (b) both involve distance and velocity. However, each required a different method or approach. Describe the relationship between distance and velocity, as well as the derivative and area under a curve.

Compute without a calculator

4-77.

If $f$ is an odd function and $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ , evaluate: 4-77 HW eTool Homework Help ✎
1. $g(–3)$
2. $2 · g(–7)$
3. $g(9) + g(–9)$
If $f$ is an even function and $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ , evaluate:
1. $g(–3)$
2. $2 · g(–7)$
3. $g(9) + g(–9)$

Continuous linear function labeled, f of t, starting at the origin, turning horizontal at the point (4, comma 4), continuing to the right.

Calculus Third Edition

Indefinite Integrals