# 1.4 Integration formulas and the net change theorem  (Page 4/8)

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## Integrating an odd function

Evaluate the definite integral of the odd function $-5\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval $\left[\text{−}\pi ,\pi \right].$

The graph is shown in [link] . We can see the symmetry about the origin by the positive area above the x -axis over $\left[\text{−}\pi ,0\right],$ and the negative area below the x -axis over $\left[0,\pi \right].$ We have

$\begin{array}{ll}{\int }_{\text{−}\pi }^{\pi }-5\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}xdx\hfill & =-5\left(\text{−}\text{cos}\phantom{\rule{0.1em}{0ex}}x\right){|}_{\text{−}\pi }^{\pi }\hfill \\ \\ \\ & =5\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x{|}_{\text{−}\pi }^{\pi }\hfill \\ & =\left[5\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\pi \right]-\left[5\phantom{\rule{0.1em}{0ex}}\text{cos}\left(\text{−}\pi \right)\right]\hfill \\ & =-5-\left(-5\right)\hfill \\ & =0.\hfill \end{array}$

Integrate the function ${\int }_{-2}^{2}{x}^{4}dx.$

$\frac{64}{5}$

## Key concepts

• The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.
• The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x -axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.

## Key equations

• Net Change Theorem
$F\left(b\right)=F\left(a\right)+{\int }_{a}^{b}F\text{'}\left(x\right)dx$ or ${\int }_{a}^{b}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right)$

Use basic integration formulas to compute the following antiderivatives.

$\int \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx$

$\int \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx=\int {x}^{1\text{/}2}dx-\int {x}^{-1\text{/}2}dx=\frac{2}{3}{x}^{3\text{/}2}+{C}_{1}-2{x}^{1\text{/}2}+{C}_{2}=\frac{2}{3}{x}^{3\text{/}2}-2{x}^{1\text{/}2}+C$

$\int \left({e}^{2x}-\frac{1}{2}{e}^{x\text{/}2}\right)dx$

$\int \frac{dx}{2x}$

$\int \frac{dx}{2x}=\frac{1}{2}\text{ln}|x|+C$

$\int \frac{x-1}{{x}^{2}}dx$

${\int }_{0}^{\pi }\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x-\text{cos}\phantom{\rule{0.1em}{0ex}}x\right)dx$

${\int }_{0}^{\pi }\text{sin}\phantom{\rule{0.1em}{0ex}}xdx-{\int }_{0}^{\pi }\text{cos}\phantom{\rule{0.1em}{0ex}}xdx=\text{−}\text{cos}\phantom{\rule{0.1em}{0ex}}x{|}_{0}^{\pi }-\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right){|}_{0}^{\pi }=\left(\text{−}\left(-1\right)+1\right)-\left(0-0\right)=2$

${\int }_{0}^{\pi \text{/}2}\left(x-\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)dx$

Write an integral that expresses the increase in the perimeter $P\left(s\right)$ of a square when its side length s increases from 2 units to 4 units and evaluate the integral.

$P\left(s\right)=4s,$ so $\frac{dP}{ds}=4$ and ${\int }_{2}^{4}4ds=8.$

Write an integral that quantifies the change in the area $A\left(s\right)={s}^{2}$ of a square when the side length doubles from S units to 2 S units and evaluate the integral.

A regular N -gon (an N -sided polygon with sides that have equal length s , such as a pentagon or hexagon) has perimeter Ns . Write an integral that expresses the increase in perimeter of a regular N -gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

${\int }_{1}^{2}Nds=N$

The area of a regular pentagon with side length $a>0$ is pa 2 with $p=\frac{1}{4}\sqrt{5+\sqrt{5+2\sqrt{5}}}.$ The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area.

A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?

With p as in the previous exercise, each of the 12 pentagons increases in area from 2 p to 4 p units so the net increase in the area of the dodecahedron is 36 p units.

An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2 a units?

Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2 s units and evaluate the integral.

$18{s}^{2}=6{\int }_{s}^{2s}2xdx$

Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2 s units and evaluate the integral.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
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Abigail
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NANO
what is fullerene does it is used to make bukky balls
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
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