# 5.4 Integration formulas and the net change theorem  (Page 3/8)

 Page 3 / 8

## Chapter opener: iceboats

As we saw at the beginning of the chapter, top iceboat racers ( [link] ) can attain speeds of up to five times the wind speed. Andrew is an intermediate iceboater, though, so he attains speeds equal to only twice the wind speed. Suppose Andrew takes his iceboat out one morning when a light 5-mph breeze has been blowing all morning. As Andrew gets his iceboat set up, though, the wind begins to pick up. During his first half hour of iceboating, the wind speed increases according to the function $v\left(t\right)=20t+5.$ For the second half hour of Andrew’s outing, the wind remains steady at 15 mph. In other words, the wind speed is given by

$v\left(t\right)=\left\{\begin{array}{lll}20t+5\hfill & \text{for}\hfill & 0\le t\le \frac{1}{2}\hfill \\ 15\hfill & \text{for}\hfill & \frac{1}{2}\le t\le 1.\hfill \end{array}$

Recalling that Andrew’s iceboat travels at twice the wind speed, and assuming he moves in a straight line away from his starting point, how far is Andrew from his starting point after 1 hour?

To figure out how far Andrew has traveled, we need to integrate his velocity, which is twice the wind speed. Then

Distance $={\int }_{0}^{1}2v\left(t\right)dt.$

Substituting the expressions we were given for $v\left(t\right),$ we get

$\begin{array}{cc}{\int }_{0}^{1}2v\left(t\right)dt\hfill & ={\int }_{0}^{1\text{/}2}2v\left(t\right)dt+{\int }_{1\text{/}2}^{1}2v\left(t\right)dt\hfill \\ & ={\int }_{0}^{1\text{/}2}2\left(20t+5\right)dt+{\int }_{1\text{/}3}^{1}2\left(15\right)dt\hfill \\ & ={\int }_{0}^{1\text{/}2}\left(40t+10\right)dt+{\int }_{1\text{/}2}^{1}30dt\hfill \\ & =\left[20{t}^{2}+10t\right]{|}_{0}^{1\text{/}2}+\left[30t\right]{|}_{1\text{/}2}^{1}\hfill \\ & =\left(\frac{20}{4}+5\right)-0+\left(30-15\right)\hfill \\ & =25.\hfill \end{array}$

Andrew is 25 mi from his starting point after 1 hour.

Suppose that, instead of remaining steady during the second half hour of Andrew’s outing, the wind starts to die down according to the function $v\left(t\right)=-10t+15.$ In other words, the wind speed is given by

$v\left(t\right)=\left\{\begin{array}{lll}20t+5\hfill & \text{for}\hfill & 0\le t\le \frac{1}{2}\hfill \\ -10t+15\hfill & \text{for}\hfill & \frac{1}{2}\le t\le 1.\hfill \end{array}$

Under these conditions, how far from his starting point is Andrew after 1 hour?

17.5 mi

## Integrating even and odd functions

We saw in Functions and Graphs that an even function    is a function in which $f\left(\text{−}x\right)=f\left(x\right)$ for all x in the domain—that is, the graph of the curve is unchanged when x is replaced with − x . The graphs of even functions are symmetric about the y -axis. An odd function    is one in which $f\left(\text{−}x\right)=\text{−}f\left(x\right)$ for all x in the domain, and the graph of the function is symmetric about the origin.

Integrals of even functions, when the limits of integration are from − a to a , involve two equal areas, because they are symmetric about the y -axis. Integrals of odd functions, when the limits of integration are similarly $\left[\text{−}a,a\right],$ evaluate to zero because the areas above and below the x -axis are equal.

## Rule: integrals of even and odd functions

For continuous even functions such that $f\left(\text{−}x\right)=f\left(x\right),$

${\int }_{\text{−}a}^{a}f\left(x\right)dx=2{\int }_{0}^{a}f\left(x\right)dx.$

For continuous odd functions such that $f\left(\text{−}x\right)=\text{−}f\left(x\right),$

${\int }_{\text{−}a}^{a}f\left(x\right)dx=0.$

## Integrating an even function

Integrate the even function ${\int }_{-2}^{2}\left(3{x}^{8}-2\right)dx$ and verify that the integration formula for even functions holds.

The symmetry appears in the graphs in [link] . Graph (a) shows the region below the curve and above the x -axis. We have to zoom in to this graph by a huge amount to see the region. Graph (b) shows the region above the curve and below the x -axis. The signed area of this region is negative. Both views illustrate the symmetry about the y -axis of an even function. We have

$\begin{array}{ll}{\int }_{-2}^{2}\left(3{x}^{8}-2\right)dx\hfill & =\left(\frac{{x}^{9}}{3}-2x\right){|}_{-2}^{2}\hfill \\ \\ \\ & =\left[\frac{{\left(2\right)}^{9}}{3}-2\left(2\right)\right]-\left[\frac{{\left(-2\right)}^{9}}{3}-2\left(-2\right)\right]\hfill \\ & =\left(\frac{512}{3}-4\right)-\left(-\frac{512}{3}+4\right)\hfill \\ & =\frac{1000}{3}.\hfill \end{array}$

To verify the integration formula for even functions, we can calculate the integral from 0 to 2 and double it, then check to make sure we get the same answer.

$\begin{array}{ll}{\int }_{0}^{2}\left(3{x}^{8}-2\right)dx\hfill & =\left(\frac{{x}^{9}}{3}-2x\right){|}_{0}^{2}\hfill \\ \\ & =\frac{512}{3}-4\hfill \\ & =\frac{500}{3}\hfill \end{array}$

Since $2·\frac{500}{3}=\frac{1000}{3},$ we have verified the formula for even functions in this particular example.

what is mathematics
logical usage of numbers
Leo
thanks
Henry
you welcome
Leo
what's career can one specialize in by doing pure maths
Lucy
Lucy Omollo...... The World is Yours by specializing in pure math. Analytics, Financial engineering ,programming, education, combinatorial mathematics, Game Theory. your skill-set will be like water a necessary element of survival.
David
***onetonline.org/find/descriptor/result/1.A.1.c.1
Bruce
mathematics seems to be anthropocentric deductive reasoning and a little high order logic. I only say this because I can only find two things going on which is infinitely smaller than 0 and anything over 1
David
More comprehensive list here: ***onetonline.org/find/descriptor/result/1.A.1.c.1?a=1
Bruce
so how can we differentiate inductive reasoning and deductive reasoning
Henry
thanks very much Mr David
Henry
hi everyone
Sabir
is there anyone who can guide me in learning the mathematics easily
Sabir
Hi Sabir first step of learning mathematics is by falling in love with it and secondly, watch videos on simple algebra then read and solved problems on it
Leo
yes sabir just do every time practice that is the solution
Henry
it will be work over to you ,u know how mind work ,it prossed the information easily when u are practising regularly
Henry
in calculas,does a self inverse function exist
Lucy
I'm lost in all functions need help
Jonathan
hello i need help in rate of change
Moises
***questioncove.com/invite/QzOQGp
Bruce
hello my name is Charles Christian
Hello Charles
Jianna
Hi! I am Dante
Dante
Hi! I'm ashwini
ashwini
halĺo
Roben
Hi
Leo
hello leo
Agboke
can anyone prove why AU(BnC)=(AUB)n(AUC)
Agboke
this one it can't be proven these are assumption
Henry
hello agboke there is no proof for such
Leo
Hi
hi this is wasim
wasim
can anybody put me through flowchart and algorithm here
Agboke
Leo
Luis
music while you math
Luis
dy/dx= 1-cos4x/sin4x
what is the derivatives of 1-cos4x/sin4x
Alma
what is the derivate of Sec2x
Johar
d/dx(sec(2 x)) = 2 tan(2 x) sec(2 x)
AYAN
who knows more about mathematical induction?
Agboke
who know anything about the whole calculus thing 🤔 its killing me 😶
matbakh
Yes
What is the integral of 2xe^x
Okay, so the chat works when I use the browser but why does the chat not work in the app?
Bruce
what is lim x=0 Tan 5x/x
integration of 3 cos^2 x - 1
keerthi
Test
Bruce
hillo
Naseer
3/4 sin2x+ 1/2 x+ C
differentiate f(x)=1/2√x
1/-2x^2
chrispine
some symbols are not shown property
Mr
f(x)=1/2âx
Mr
how to solve e for slope eequations
I am willinv to know this. Though i know little bit.
m = y2 -y1 / x2 - x1
Reymund
explain mercuries theorem?
limit of f(x) sec x - 1 /1 - sec x while x is approaching 0
hi is someone in there
hello
This
hi
Agboke
please can anyone recommend a very good maths text book for me? 100 level student
Agboke
for algebra?
Elysha
Hi
Gajji
I don't know all these as i just got in
Agboke
I trying to remember haven't done this in years
Chavon
Paul
okay thanks
Agboke
hi
Rabbit
yes
Nelson
hi sir
Rabbit
Hello
hassan
Beta function
Allu
what is xsin (1/x)
1
The denominator of a fraction is 5 more than the numerator. If half the numerator plus one is added to both terms of the fractions, the resulting fraction would be 5/6. Find the original fraction.
Conney
what is (x+1/x)
Agboke
proof that(A u B)=A' n B'
Do you mean (A u B)' = A' n B'?
WC
use the De Morgan's laws
Marcray
f(x)=3+|x|-x^4. Is this function even, odd or neither?
how to find a limit algebraically
Follow the rules of limit and evaluate
what is Cale nation
its probably a football leage in kentucky
Luis