5.4 Integration formulas and the net change theorem

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• Apply the basic integration formulas.
• Explain the significance of the net change theorem.
• Use the net change theorem to solve applied problems.
• Apply the integrals of odd and even functions.

In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind. A definite integral is either a number (when the limits of integration are constants) or a single function (when one or both of the limits of integration are variables). An indefinite integral represents a family of functions, all of which differ by a constant. As you become more familiar with integration, you will get a feel for when to use definite integrals and when to use indefinite integrals. You will naturally select the correct approach for a given problem without thinking too much about it. However, until these concepts are cemented in your mind, think carefully about whether you need a definite integral or an indefinite integral and make sure you are using the proper notation based on your choice.

Basic integration formulas

Recall the integration formulas given in [link] and the rule on properties of definite integrals. Let’s look at a few examples of how to apply these rules.

Integrating a function using the power rule

Use the power rule to integrate the function ${\int }_{1}^{4}\sqrt{t}\left(1+t\right)dt.$

The first step is to rewrite the function and simplify it so we can apply the power rule:

$\begin{array}{cc}{\int }_{1}^{4}\sqrt{t}\left(1+t\right)dt\hfill & ={\int }_{1}^{4}{t}^{1\text{/}2}\left(1+t\right)dt\hfill \\ \\ & ={\int }_{1}^{4}\left({t}^{1\text{/}2}+{t}^{3\text{/}2}\right)dt.\hfill \end{array}$

Now apply the power rule:

$\begin{array}{cc}{\int }_{1}^{4}\left({t}^{1\text{/}2}+{t}^{3\text{/}2}\right)dt\hfill & ={\left(\frac{2}{3}{t}^{3\text{/}2}+\frac{2}{5}{t}^{5\text{/}2}\right)|}_{1}^{4}\hfill \\ & =\left[\frac{2}{3}{\left(4\right)}^{3\text{/}2}+\frac{2}{5}{\left(4\right)}^{5\text{/}2}\right]-\left[\frac{2}{3}{\left(1\right)}^{3\text{/}2}+\frac{2}{5}{\left(1\right)}^{5\text{/}2}\right]\hfill \\ & =\frac{256}{15}.\hfill \end{array}$

Find the definite integral of $f\left(x\right)={x}^{2}-3x$ over the interval $\left[1,3\right].$

$-\frac{10}{3}$

The net change theorem

The net change theorem    considers the integral of a rate of change . It says that when a quantity changes, the new value equals the initial value plus the integral of the rate of change of that quantity. The formula can be expressed in two ways. The second is more familiar; it is simply the definite integral.

Net change theorem

The new value of a changing quantity equals the initial value plus the integral of the rate of change:

$\begin{array}{}\\ \\ F\left(b\right)=F\left(a\right)+{\int }_{a}^{b}F\text{'}\left(x\right)dx\hfill \\ \hfill \text{or}\hfill \\ {\int }_{a}^{b}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right).\hfill \end{array}$

Subtracting $F\left(a\right)$ from both sides of the first equation yields the second equation. Since they are equivalent formulas, which one we use depends on the application.

The significance of the net change theorem lies in the results. Net change can be applied to area, distance, and volume, to name only a few applications. Net change accounts for negative quantities automatically without having to write more than one integral. To illustrate, let’s apply the net change theorem to a velocity function in which the result is displacement .

We looked at a simple example of this in The Definite Integral . Suppose a car is moving due north (the positive direction) at 40 mph between 2 p.m. and 4 p.m., then the car moves south at 30 mph between 4 p.m. and 5 p.m. We can graph this motion as shown in [link] .

I don't understand the set builder nototation like in this case they've said numbers greater than 1 but less than 5 is there a specific way of reading {x|1<x<5} this because I can't really understand
x is equivalent also us 1...
Jogimar
a < x < b means x is between a and b which implies x is greater than a but less than b.
Bruce
alright thanks a lot I get it now
Ivwananji
I'm trying to test this to see if I am able to send and receive messages.
👍
Jogimar
It's possible
joseph
yes
Waidus
Anyone with pdf tutorial or video should help
joseph
maybe.
Jogimar
hellow gays. is it possible to ask a mathematical question
Michael
What is the derivative of 3x to the negative 3
-3x^-4
Mugen
that's wrong, its -9x^-3
Mugen
Or rather kind of the combination of the two: -9x^(-4) I think. :)
Csaba
-9x-4 (X raised power negative 4=
Simon
-9x-³
Jon
I have 50 rupies I spend as below Spend remain 20 30 15 15 09 06 06 00 ----- ------- 50 51 why one more
Pls Help..... if f(x) =3x+2 what is the value of x whose image is 5
f(x) = 5 = 3x +2 x = 1
x=1
Bra
can anyone teach me how to use synthetic in problem solving
Mark
x=1
Mac
can someone solve Y=2x² + 3 using first principle of differentiation
ans 2
Emmanuel
Yes ☺️ Y=2x+3 will be {2(x+h)+ 3 -(2x+3)}/h where the 2x's and 3's cancel on opening the brackets. Then from (2h/h)=2 since we have no h for the limit that tends to zero, I guess that is it....
Philip
Correct
Mohamed
thank you Philip Kotia
Rachael
Welcome
Philip
g(x)=8-4x sqrt of 3 + 2x sqrt of 8 what is the answer?
Sheila
I have no idea what these symbols mean can it be explained in English words
which symbols
John
How to solve lim x squared two=4
why constant is zero
Rate of change of a constant is zero because no change occurred
Highsaint
What is the derivative of sin(x + y)=x + y ?
_1
Abhay
How? can you please show the solution?
Frendick
neg 1?
Frendick
find x^2 + cot (xy) =0 Dy/dx
Continuos and discontinous fuctions
integrate dx/(1+x) root 1-x square
Put 1-x and u^2
Ashwini
hi
telugu
d/dxsinh(2xsquare-6x+4)
how
odofin
how to do trinomial factoring?
Give a trinomial to factor and I'll show you how
Bruce
i dont know how to do that either. but i really wants to know know how
Bern
Do either of you have a trinomial to present that needs factoring?
Bruce
5x^2+11x+2 and 2x^2+7x-4
Samantha
Thanks, so I'll show you how to do the 1st one and then you can try to do the second one. The method I show you is a general method you can use to factor ANY factorable polynomial.
Bruce
ok
Samantha
I have to link you to a document on how to do it since this chat does not have LaTeX
Bruce
Give me a few
Bruce
Sorry to keep you waiting. Here it is: ***mathcha.io/editor/ZvZkJUdrHpVHXwhe7
Bruce
You are welcome to ask questions if you have them. Good luck
Bruce
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Lemisa
If you want
Bruce
I don't see an option for photo
Lemisa
Do you know Math?
Lemisa
You have to post it as a link
Bruce
Can you tell me how to. Post?
Lemisa
Yes, you can upload your photo on imgur and post the link here
Bruce
thanks it was helpful
Samantha
I'm glad that was able to help you. Seriously, if you have any questions, please ask. That is not easy to understand at first so I anticipate you will have questions. It would be best to convince that you understood to attempt factoring the second trinomial you posted.
Bruce