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Applications of Integration

In mathematics and physics integration is a process of finding an area or volume by tracing out its boundary.. Integration is a fundamental tool of calculus and its various applications in science and engineering. Integration is a key concept in several mathematical disciplines, including differential calculus, complex analysis, topology and functional analysis. A: In mathematics, the integral of a function of a variable takes values in a function's range. When the function is evaluated at a particular value of its independent variable, the result of the integral is the value of the function at that point. The integrals of the elementary functions, such as the sine and cosine functions, are defined using the fundamental theorem of calculus, which in turn is based on the differential calculus. The elementary functions are used to define the more complicated functions by integration, which is the reverse process of differentiation. In calculus, the integrals of elementary functions which are defined by simple rules of the form are known as elementary functions. To take one example, the integral of the cosine function is For example, the integral of the secant function is In many applications, the integrals of any given function "f" of a real variable can be written in the form where "C" is a constant of integration known as the constant of integration. The constant of integration is often omitted in notation when the context is clear, as in The constant of integration may be omitted when the context is clear; for example, In the case of polynomial functions, the constant of integration is sometimes called the constant of integration or leading coefficient. For example, the integral of the polynomial function can be written as The constant of integration is an integer, and the integral of a polynomial function contains the nth degree term, or leading coefficient. In this case, the constant of integration is the leading coefficient. The remaining terms are known as the constant term. The Laplace transform and Fourier transform are inverse operations to differentiation and integration. The concept of the integral was developed by Isaac Newton in his "Principia", in which he defined the area under a curve as the limit of sums of infinitesimal areas of rectangles fitted snugly under the curve. Newton's method is the method of exhaustion, in which an area is approximated by successively adding up a series of rectangles whose widths are in a fixed ratio to the area sought. Here is an example with a rectangle of width 1 and length "x" and of height "y" (written in terms of the trigonometric functions): This is a simple example of a method of exhaustion, in which the width of each rectangle "varies in proportion to the area to be approximated". The method of limits was developed by Carl Friedrich Gauss and Jakob Bernoulli at the end of the 17th century. Here is an example of the method of limits, again with a rectangle of width "x" and length "y" and of height "z". The method of least squares is an example of the method of variation of variables. The basic idea is to vary one of the parameters of the problem, for example, the width of the rectangle. The approximate area of the rectangle is then calculated and compared with the exact area. The first step is to find the approximate area of the rectangle. This can be done by first finding the area of a smaller rectangle with the same height and width, but a smaller width. Then, the ratio of the area of this smaller rectangle to the area of the larger rectangle is used to find the width of the larger rectangle. The method of limits is used to find the approximate area of the larger rectangle. In the case of a curve, the concept of an integral can be extended to include areas under the graph of the function. For example, the area under the graph of the function is the integral of the function from a limit as "x" approaches 0 to infinity. The area under the curve between the limits 0 and 1 is the limit as "x" approaches 1 from 0, and is given by the definite integral of the function from 0 to 1. The area under the curve between the limits 1 and 1 is the limit as "x" approaches 1 from 1, and is given by the indefinite integral of the function from 1 to 1. The idea of integrating the area between a curve and the x-axis is also used to find the area between two curves. For example, the area between the graph of the function and the line "y" = "x" is given by the definite integral of the function from 0 to 1. The area between the graph of the function and the line "y" = "x" ? "a" is given by the indefinite integral of

Areas Between Curves

259 Practice Problems
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03:12
Calculus for Scientists and Engineers: Early Transcendental

Determine the area of the shaded region bounded by the curve $x^{2}=y^{4}\left(1-y^{3}\right)$ (see figure). (FIGURE CANNOT COPY)

Applications of Integration
Regions Between Curves
John Peter
06:09
Calculus for Scientists and Engineers: Early Transcendental

Either method Use the most efficient strategy for computing the area of the following regions.
The region in the first quadrant bounded by $y=x^{-1}, y=4 x,$ and$$y=x / 4$$

Applications of Integration
Regions Between Curves
David Marsella
08:06
Calculus for Scientists and Engineers: Early Transcendental

Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves
and the region in question.
The region between the line $y=x$ and the curve $y=2 x \sqrt{1-x^{2}}$ in the first quadrant

Applications of Integration
Regions Between Curves
David Marsella

Volumes

390 Practice Problems
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06:10
Chemistry: Introducing Inorganic, Organic and Physical Chemistry

A vessel of volume $50.0 \mathrm{dm}^{3}$ contains $2.50 \mathrm{mol}$ of argon and $1.20 \mathrm{mol}$ of nitrogen at $273.15 \mathrm{K}$
(i) Calculate the partial pressure in bar of each gas.
(ii) Calculate the total pressure in bar.
(iii) How many additional moles of nitrogen must be pumped into the vessel in order to raise the pressure to 5 bar? (Sections $8.2 \text { and } 8.3)$

Gases
Lottie Adams
01:58
Chemistry: Introducing Inorganic, Organic and Physical Chemistry

A sample of gas has a volume of $346 \mathrm{cm}^{3}$ at $25^{\circ} \mathrm{C}$ when the pressure is 1.00 atm. What volume will it occupy if the conditions are changed to $35^{\circ} \mathrm{C}$ and 1.25 atm? (Section 8.2)

Gases
Nicole Smina
02:13
Chemistry: Introducing Inorganic, Organic and Physical Chemistry

A sealed flask holds $10 \mathrm{dm}^{3}$ of gas. What is this volume in
(a) $\left.\mathrm{cm}^{3},(\mathrm{b}) \mathrm{m}^{3}, \text { (c) litres? (Section } 1.2\right)$

Fundamentals
Nicole Smina

Volumes by Cylindrical Shells

135 Practice Problems
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05:19
Calculus for Scientists and Engineers: Early Transcendental

Change of variables Suppose $f(x)>0$ for all $x$ and $\int_{0}^{4} f(x) d x=10 .$ Let $R$ be the region in the first quadrant bounded by the coordinate axes, $y=f\left(x^{2}\right),$ and $x=2 .$ Find the volume of the solid generated by revolving $R$ around the $y$ -axis.

Applications of Integration
Volume by Shells
Dorcas Attuabea Addo
08:58
Calculus for Scientists and Engineers: Early Transcendental

Water in a bowl A hemispherical bowl of radius 8 inches is filled to a depth of $h$ inches, where $0 \leq h \leq 8(h=0$ corresponds to an empty bowl). Use the shell method to find the volume of water in the bowl as a function of $h$. (Check the special cases $h=0$ and $h=8 .)$

Applications of Integration
Volume by Shells
Dorcas Attuabea Addo
02:24
Calculus for Scientists and Engineers: Early Transcendental

Find the volume of the following solids using the method of your choice.
The solid formed when the region bounded by $y=\sqrt{x},$ the $x$ -axis, and $x=4$ is revolved about the $x$ -axis

Applications of Integration
Volume by Shells
Dorcas Attuabea Addo

Work

380 Practice Problems
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06:33
Physical Chemistry

How much work is required to bring two protons from an infinite distance of separation to $0.1 \mathrm{nm} ?$ Calculate the answer in joules using the protonic charge $1.602 \times 10^{-19} \mathrm{C}$. What is the work in $\mathrm{kJ} \mathrm{mol}^{-1}$ for a mole of proton pairs?

Electrochemical Equilibrium
Sharfa Farzandh
02:15
Physical Chemistry

How high can a person (assume a weight of $70 \mathrm{kg}$ ) climb on one ounce of chocolate, if the heat of combustion $(628 \mathrm{kJ}) \mathrm{can}$ be converted completely into work of vertical displacement?

First Law of Thermodynamics
Lottie Adams
09:45
Fundamentals of Thermodynamics

A steam turbine has an inlet of $2 \mathrm{kg} / \mathrm{s}$ water at 1000 $\mathrm{kPa}$ and $350^{\circ} \mathrm{C}$ with velocity of $15 \mathrm{m} / \mathrm{s}$. The exit is at $100 \mathrm{kPa}, x=1,$ and very low velocity. Find the specific work and the power produced.

First Law Analysis for a Control Volume
Sanu Kumar

Average Value of a Function

86 Practice Problems
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01:20
Calculus: Early Transcendental Functions

Find the average value of the function on the given interval.
$f(x)=x^{2}-1,[1,3]$

Integration
The Fundamental Theorem of Calculus
Ernest Castorena
04:52
Calculus: Early Transcendental Functions

Compute the average value of the function on the given interval.
$$f(x)=2 x+1,[0,4]$$

Integration
The Definite Integral
Ishita J.
08:11
University Calculus: Early Transcendentals

Average value If $f$ is continuous, the average value of the polar coordinate $r$ over the curve $r=f(\theta), \alpha \leq \theta \leq \beta,$ with respect to $\theta$ is given by the formula
$$r_{\mathrm{av}}=\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta} f(\theta) d \theta$$
Use this formula to find the average value of $r$ with respect to $\theta$ over the following curves $(a>0)$
a. The cardioid $r=a(1-\cos \theta)$
b. The circle $r=a$
c. The circle $r=a \cos \theta, \quad-\pi / 2 \leq \theta \leq \pi / 2$

Parametric Equations and Polar Coordinates
Areas and Lengths in Polar Coordinates
Carlos Pinilla

Arc Length

242 Practice Problems
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05:47
Calculus for Scientists and Engineers: Early Transcendental

Consider a particle that moves in a plane according to the equations $x=\sin t^{2}$ and $y=\cos t^{2}$ with a starting position (0,1) at $t=0$
a. Describe the path of the particle, including the time required to return to the starting position.
b. What is the length of the path in part (a)?
c. Describe how the motion of this particle differs from the motion described by the equations $x=\sin t$ and $y=\cos t$
d. Now consider the motion described by $x=\sin t^{n}$ and $y=\cos t^{n}$ where $n$ is a positive integer. Describe the path of the particle, including the time required to return to the starting position.
e. What is the length of the path in part (d) for any positive integer $n ?$
f. If you were watching a race on a circular path between two runners, one moving according to $x=\sin t$ and $y=\cos t$ and one according to $x=\sin t^{2}$ and $y=\cos t^{2},$ who would win and when would one runner pass the other?

Vectors and Vector-Valued Functions
Length of Curves
Joseph Liao
02:34
Calculus for Scientists and Engineers: Early Transcendental

Consider the spiral $r=4 \theta,$ for $\theta \geq 0$
a. Use a trigonometric substitution to find the length of the spiral, for $0 \leq \theta \leq \sqrt{8}$
b. Find $L(\theta),$ the length of the spiral on the interval $[0, \theta],$ for any $\theta \geq 0$
c. Show that $L^{\prime}(\theta)>0 .$ Is $L^{\prime \prime}(\theta)$ positive or negative? Interpret your answers.

Vectors and Vector-Valued Functions
Length of Curves
Joseph Liao
03:02
Calculus for Scientists and Engineers: Early Transcendental

Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter.
$$\mathbf{r}(t)=\left\langle t^{2}, 2 t^{2}, 4 t^{2}\right\rangle, \text { for } 1 \leq t \leq 4$$

Vectors and Vector-Valued Functions
Length of Curves
Joseph Liao

Area of a Surface of Revolution

136 Practice Problems
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08:29
Engineering Mechanics: Statics and Dynamics

Determine the radii of gyration $k_{x}$ and $k_{y}$ for the solid formed by revolving the shaded area about the $y$ axis. The density of the material is $\rho$

Three-Dimensional Kinetics of a Rigid Body
Shoukat Ali
06:55
Engineering Mechanics: Statics and Dynamics

Determine the moment of inertia $I_{y}$ of the object formed by revolving the shaded area about the line $x=5 \mathrm{ft}$ Express the result in terms of the density of the material, $\rho$

Three-Dimensional Kinetics of a Rigid Body
Shoukat Ali
04:17
Engineering Mechanics: Statics and Dynamics

Determine the product of inertia $I_{x y}$ of the object formed by revolving the shaded area about the line $x=5 \mathrm{ft}$ Express the result in terms of the density of the material, $\rho$

Three-Dimensional Kinetics of a Rigid Body
Shoukat Ali

Probability

430 Practice Problems
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00:43
Elements of Physical Chemistry

At low temperatures a substituted 1,2 -dichloroethane molecule can adopt the three conformations $(2),(3),$ and (4) with different probabilities. Suppose that the dipole moment of each $\mathrm{C}-\mathrm{C}$ 1 bond is 1.50 D. Calculate the mean dipole moment of the molecule when (a) all three conformations are equally likely, (b) only conformation (2) occurs, (c) the three conformations occur with probabilities in the ratio 2: 1: 1 and $(d) 1: 2: 2$.

Molecular interactions
Ayushi Sambyal
04:58
21st Century Astronomy

Neutral hydrogen emits radiation at a radio wavelength of $21 \mathrm{cm}$ when an atom drops from a higher-energy spin state to a lower-energy spin state. On average, each atom remains in the higher energy state for 11 million years $\left(3.5 \times 10^{14}$ seconds) \right.
a. What is the probability that any given atom will make the transition in 1 second?
b. If there are $6 \times 10^{59}$ atoms of neutral hydrogen in a $500-M_{\text {sun }}$ cloud, how many photons of 21 -cm radiation will the cloud emit each second?
c. How does this number compare with the $1.8 \times 10^{45}$ photons emitted each second by a solar-type star?

The Interstellar Medium and Star Formation
Sarah Mccrumb
03:29
Chemistry

Referring to Figure $18.1,$ we see that the probability of finding all 100 molecules in the same flask is $8 \times 10^{-31}$. Assuming that the age of the universe is 13 billion years, calculate the time in seconds during which this event can be observed.

Entropy, Free Energy, and Equilibrium
Luke Monroe

Find Volume by: Method of Cross Sections

27 Practice Problems
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03:41
Calculus: Early Transcendental Functions

The base of a solid $V$ is the region bounded by $y=x^{2}$ and $y=2-x^{2} .$ Find the volume if $V$ has (a) square cross sections, (b) semicircular cross sections and (c) equilateral triangle cross sections perpendicular to the $x$ -axis.

Applications of the Definite Integral
volume: Slicing, Disks and Washers
Gregory Higby
01:39
Calculus: Early Transcendental Functions

A pottery jar has circular cross sections of radius $4+\sin \frac{x}{2}$ inches for $0 \leq x \leq 2 \pi .$ Sketch a picture of the jar and compute its volume.

Applications of the Definite Integral
volume: Slicing, Disks and Washers
Robert Leedy
13:23
Calculus: Early Transcendentals

Solids from integrals Sketch a solid of revolution whose volume by the disk method is given by the following integrals. Indicate the function that generates the solid. Solutions are not unique.
$$\text { a. } \int_{0}^{\pi} \pi \sin ^{2} x d x$$
$$\text { b. } \int_{0}^{2} \pi\left(x^{2}+2 x+1\right) d x$$

Applications of Integration
Volume by Slicing
Tp Sarathy

Find Volume by: Method of Disks (rings)

20 Practice Problems
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01:42
Calculus: Early Transcendental Functions

Suppose that the square consisting of all points $(x, y)$ with $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$ is revolved about the $y$ -axis. Show that the volume of the resulting solid is $2 \pi$

Applications of the Definite Integral
volume: Slicing, Disks and Washers
Gregory Higby
02:29
Calculus of a Single Variable

The integral represents the volume of a solid. Describe the solid.
$$\pi \int_{0}^{\pi / 2} \sin ^{2} x d x$$

Applications of Integration
Volume: The Disk Method
Gregory Higby
01:13
Calculus of a Single Variable

Find the volume generated by rotating the given region about the specified line.
$$R_{1} \text { about } x=0$$

Applications of Integration
Volume: The Disk Method
Gregory Higby

Find Volume by: Method of Cylinders

19 Practice Problems
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02:51
Calculus: Early Transcendental Functions

Sketch the region, draw in a typical shell, identify the radius and height of each shell and compute the volume.
The region bounded by $y=x^{2}$ and the $x$ -axis, $-1 \leq x \leq 1$ revolved about $x=2$.

Applications of the Definite Integral
Volumes by Cylindrical Shells
Abigail Martyr
02:10
Thomas Calculus

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines in Exercises
$x=2 y-y^{2}, \quad x=0$

Applications Of Definite Integrals
Volumes by Cylindrical Shells
Kyler Gray
02:31
Calculus Volume 2

Consider the function $y=f(x),$ which decreases from $f(0)=b$ to $f(1)=0 .$ Set up the integrals for
determining the volume, using both the shell method and the disk method, of the solid generated when this region, with $x=0$ and $y=0, \quad$ is rotated around the $y$ -axis. Prove that both methods approximate the same volume. Which method is easier to apply? (Hint: since $f(x)$ is one-to-one, there exists an inverse $f^{-1}(y) . )$

Applications of Integration
Volumes of Revolution: Cylindrical Shells
Ad F

Center of Mass (Centroids)

123 Practice Problems
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04:00
Engineering Mechanics: Statics and Dynamics

Locate the centroid $\bar{y}$ of the shaded area.

Center of Gravity and Centroid
Khoobchandra Agrawal
03:47
Engineering Mechanics: Statics and Dynamics

Locate the centroid $\bar{x}$ of the shaded area.

Center of Gravity and Centroid
Khoobchandra Agrawal
04:04
Engineering Mechanics: Statics and Dynamics

Locate the centroid $\bar{y}$ of the shaded area.

Center of Gravity and Centroid
Khoobchandra Agrawal

Natural Log Defined as an Integral

18 Practice Problems
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01:16
Calculus: Early Transcendental Functions

Evaluate the integral.
$$\int_{0}^{1} \frac{x^{2}}{x^{3}-4} d x$$

Integration
The natural Logarithm as an Integral
Linh Vu
00:36
Algebra 2

Solve the equation. Check your answer.
$$
\ln x=-2
$$

Exponential Logarithmic Functions
Natural Logarithms
Chinmai Managoli
00:39
Algebra 2

Solve each equation.
$$
\log 5 x+3=3.7
$$

Exponential Logarithmic Functions
Natural Logarithms
Chinmai Managoli

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