What is the meaning of $+C$ ?

Indefinite and Definite Integrals

4-52.

Sketch the region represented by each integral below.
Then use geometry to write a general area function for each integral.

$\int _ { 0 } ^ { x } 2 d p$

$\int _ { 0 } ^ { x } ( 5 r + 2 ) d r$

$\int _ { 0 } ^ { x } ( \frac { 3 } { 2 } - k ) d k$

4-53.

Use your results from problem 4-52 and the formula $\int _ { c } ^ { x } f ( t ) d t = A ( x ) - A ( c )$ to evaluate the following definite integrals.

$\int _ { 3 } ^ { 8 } 2 d x$

$\int _ { - 2 } ^ { 5 } 8 d x$

$3 \int _ { 2 } ^ { 6 } ( 5 x + 2 ) d x$

$\int _ { 1 } ^ { 3 } 5 x d x + \int _ { 1 } ^ { 3 } 2 d x$

$\int _ { 4 } ^ { 10 } ( \frac { 3 } { 2 } - k ) d k$

$\int _ { 0 } ^ { - 2 } ( \frac { 3 } { 2 } - k ) d k$

4-54.

A millipede is moving along the edge of a centimeter ruler with a velocity, $v$ , given by the function $v\left(t\right) = 3t – 9$ where $v\left(t\right)$ is measured in centimeters per second.

Write and evaluate a definite integral that can be used to calculate the displacement of the millipede over the interval $t = 3$ sec to $t = 6$ . Hint: Start by sketching a graph of $y = v\left(t\right)$ .
Write and evaluate a definite integral that represents the displacement, $s\left(t\right)$ , of the millipede from $t = 3$ to any time $t = x$ . Then calculate the value of $s\left(6\right)$ .
If the millipede is standing on the $15$ centimeter mark at $t = 3$ seconds, then where will it be located at $t = 6$ seconds? How about at $t = 10$ seconds? Show and explain your process.

4-55.

Tommy is trying to determine $\int _ { 2 } ^ { 3 } f ( t ) d t$ . He already knows that $\int _ { 0 } ^ { x } f ( t ) d t = 9 x ^ { 2 } - 2 x$ . Help him calculate $\int _ { 2 } ^ { 3 } f ( t ) d t$ .

Math Notes

Definite Integrals with Variable Bounds

Definite integrals can be expressed on bounds $[a, x]$ , where $a$ is a constant and $x$ is variable. This type of definite integral results in a function called an antiderivative, which is often notated with an uppercase letter.

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

The antiderivative $F\left(x\right)$ shown above can be interpreted as an area function that will determine the area under the graph of $y = f\left(t\right)$ from $t = a$ to any value of $x$ . $F\left(x\right)$ can also be interpreted as an accumulation function, which computes the accumulated value of $f$ from $t = a$ to any value of $x$ .

An antiderivative equation depends of the integrand, $f$ , and the lower bound, $a$ . Consequently, changing the lower bound will change the antiderivative by a constant.

$\int _ { a } ^ { x } f ( t ) d t = \int _ { b } ^ { x } f ( t ) d t + C$ , where $C$ is a constant.

There are many algebraic techniques that can be used to write the equation of an antiderivative function. Some of these techniques will be explored in this course. However, it is important to know that not every function has a closed form antiderivative. That is, not every antiderivative can be found using operations from arithmetic, algebra, or trigonometry.

4-56.

Ji Hee is trying to evalute the integrals in the parts (a) through (c) below. She already knows that $\int _ { 0 } ^ { x } g ( m ) d m = \frac { 4 x + 1 } { x + 2 }$ . Help her calculate: Homework Help ✎

$2 \int _ { 0 } ^ { 3 } g ( m ) d m$
$\int _ { - 1 } ^ { 0 } g ( m ) d m$
$\int _ { - 1 } ^ { 5 } g ( m ) d m$

4-57.

Mateo and Ignacio want to calculate $\int _ { - 2 } ^ { - 5 } f ( x ) d x$ where $f$ is an even function. They know that $\int _ { 2 } ^ { 5 } f ( x ) d x = - 3$ . Mateo thinks the answer will be $–3$ while Ignacio thinks the answer will be $3$ . Who is correct? Explain. Homework Help ✎

4-58.

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

A ferry crosses the bay so that its distance (in miles) from the dock at time $t$ is $d\left(t\right)=1- \cos\left(t\right)$ . What is the velocity, $d ^\prime(t)$ , at times $t = 1$ , $π$ , and $5$ hours? Explain what concepts of calculus you applied in order to solve this problem.
When a cat chases a mouse, the cat’s velocity, measured in feet per second, is $v(t) = 3t$ . Sketch a graph and calculate the distance the cat ran in the first $5$ seconds. Explain what concepts of calculus you applied in order to solve this problem.
Both (a) and (b) involve distance and velocity. However, each required a different method or approach. Describe the relationship between distance and velocity, mentioning the derivative and area under a curve.

4-59.

While driving to work, Camille decided to keep track of the time and the distance she traveled. Taking the data that she gathered, she found the function $s$ to represent her distance as a function of time: 4-59 HW eTool Homework Help ✎

$s\left(t\right) = –100t^{3} + 150t^{2}$

If Camille’s trip took $1$ hour, how far did she drive?
What was her average velocity?
Use your graphing calculator to determine her maximum velocity. How far into the trip did she reach this speed?

First quadrant, unscaled x axis labeled, time, hours, unscaled y axis labeled, distance, miles, continuous increasing curve labeled, s of t, starting at the origin, changing from concave up to concave down at about the center.

4-60.

Change the following limit of a Riemann sum for the area under a curve into an integral expression. Then, evaluate the sum with your graphing calculator. Homework Help ✎
$\lim\limits _ { n \rightarrow \infty } \sum _ { i = n } ^ { 5 n } \frac { 1 } { n } ( ( \frac { i } { n } ) ^ { 3 } - 4 ) = \int _ { - } ^ { - } ( \underline { } ) _ { - }$

4-61.

If a function is differentiable at $x = c$ , does that guarantee that $f\left(c\right)$ exists? Explain why or why not. Homework Help ✎

4-62.

Write the equation of the tangent line to $f\left(x\right)=\sin\left(x\right)$ at $\frac { \pi } { 3 }$ . Homework Help ✎

4-63.

Differentiate. Homework Help ✎

$\frac { d } { d x } ( 7 \cdot \sqrt [ 3 ] { x } )$
$\frac { d } { d m } ( 3 m ^ { - 7 } - 7 m ^ { 3 } )$
$\frac { d } { d k } ( k ^ { 0 } )$
$\frac { d } { d t } ( ( 3 t ) ( 2 t + 5 ) )$

4-64.

What is the general antiderivative, $F$ , of each of the following functions? Homework Help ✎

$f\left(x\right)=$ $15x^4+4x-3$
$f\left(x\right)=$ $-2\cos\left(x\right)$
$f\left(x\right)=$ $-4x^3+10$

Calculus Third Edition

Definite Integrals with Variable Bounds