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Integrals

Integrals are the area of mathematics that deals with the integration of a function. The area of mathematics is concerned with the subject of the integrals themselves. Integrals are usually written on the right side of an equal sign. Integrals are also used in a variety of formulas, particularly in mathematics and physics. Integrals are usually written using a definite integral symbol (for example, ) or a function notation (for example, sin(x)dx, which is the integral of the sine function with respect to the variable x). The definite integral of a function f(x) with respect to a variable x is denoted by: or Integrals can be classified into two areas: The area called elementary calculus is devoted to the integration of elementary functions such as polynomials, rational functions, trigonometric and exponential functions. The area called advanced calculus is devoted to the integration of more complicated functions such as exponential and logarithmic functions, trigonometric and hyperbolic functions, and even functions of several variables. The remainder of this section discusses elementary functions. The indefinite integral and the definite integral are two different forms of integration. The definite integral is the standard form of integration used in calculus. The definite integral may or may not be unique; if two functions have the same definite integral, then they are equal. The indefinite integral is the integral of a function in which the limits of integration are not fixed. For example, the indefinite integral of (ex) is In the above integral, the limit of integration is infinity, but it is not specified. The indefinite integral of a function is unique if the function is continuous and its limits of integration are in the closed interval [a, b] (a closed interval is an interval with a definite end point, such as (2, 3), or [0, ?)), or if the function is differentiable with a derivative at every point of the closed interval. If it is not differentiable at every point in a closed interval, the indefinite integral of the function with respect to any variable is undefined. The indefinite integral of a function is a special case of the integral of the integral. Using the Riemann sum theorem, the indefinite integral of a function "f"("x") is given by: The fundamental theorem of calculus states that the indefinite integral of a function of a real variable over a closed interval containing the integral of the function is equal to the integral of the function of the function. The result of this theorem can be extended to the case of an unbounded interval containing the integral of the function. Where the function is defined at an open interval containing the integral, then the result of the extension is the same as the result of the theorem. The definite integral is an integral over a closed interval of the form: where "a" and "b" are real numbers and the function "f" is continuous on the closed interval. The left-hand side of the definite integral can be interpreted as the area between the curve "y" = "f"("x") and the x-axis, and the right-hand side as the area between the curve "y" = "f"("x") and the x-axis. For the function formula_3, the definite integral can be calculated as follows: The integral of a function of a single variable is the limit of a sequence of approximations of the integral of the function. The limit of a finite sequence of approximations is the same as the integral of the function. The following table shows an infinite sequence of approximations of the integral of sin ("x"): Each of these terms is a quotient of the previous term by the difference between the limits of the interval. The limit of a finite sequence of approximations, if finite, is the same as the integral of the function. The limit of an infinite sequence of approximations is a function of the limits of the sequence. The limit of the sequence is therefore the original function. As an example, let the function formula_9 be defined for all real numbers "x" as: and let us evaluate the following sequence of approximations of the integral: If we let "x" approach infinity, we get the following sequence: The limit of this infinite sequence of approximations is therefore the function formula_10. A function of two variables may be integrated over a closed interval containing the interval of integration only if one of the functions is differentiable at every point in the closed interval. The indefinite integral of a function of two variables over a closed interval containing the interval of integration is the limit of the sequence of indefinite integrals of the function of the function. The limit of an infinite sequence of indefinite integrals

Areas and Distances

160 Practice Problems
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04:52
Precalculus

A plane flew due north at 500 miles per hour for 3 hours. A second plane, starting at the same point and at the same time, flew southeast at an angle $150^{\circ}$ clockwise from due north at 435 miles per hour for 3 hours. At the end of the 3 hours, how far apart were the two planes? Round to the nearest mile.

Trigonometric Functions of Angles
The Law of Cosines
Linda Hand
01:52
University Calculus: Early Transcendentals

find the distance from the point to the plane.
$$(0,1,1), \quad 4 y+3 z=-12$$

Vectors and the Geometry of Space
Lines and Planes in Space
Linh Vu
05:15
Calculus: Early Transcendentals

Area of a triangle Find the area of the triangle with vertices on the coordinate axes at the points $(a, 0,0),(0, b, 0),$ and $(0,0, c)$ in terms of $a, b,$ and $c$

Vectors and the Geometry of Space
Cross Products
Jose Hannan

Definite Integrals

942 Practice Problems
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04:05
Calculus: Early Transcendental Functions

Prove Green's first identity in three dimensions (see exercise 43 in section 14.5 for Green's first identity in two dimensions): $\iiint_{Q} f \nabla^{2} g d V=\iint_{\partial Q} f(\nabla g) \cdot \mathbf{n} d S-\iiint_{Q}(\nabla f \cdot \nabla g) d V$ (Hint: Use the Divergence Theorem applied to $\mathbf{F}=f \nabla g$.)

Vector Calculus
The Divergence Theorem
Sriram Soundarrajan
01:04
Calculus: Early Transcendental Functions

Find the flux of $\mathbf{F}$ over $\partial Q$. $Q \quad$ is bounded by $\quad x^{2}+z^{2}=1, y=0 \quad$ and $\quad y=1$ $\mathbf{F}=\left\langle z-y^{3}, 2 y-\sin z, x^{2}-z\right\rangle$

Vector Calculus
The Divergence Theorem
Sriram Soundarrajan
03:09
Calculus: Early Transcendental Functions

Find the value of the constant $k$ to make each of the following pdf's on the interval $[0, \infty) . \text { (See exercise } 61 .)$
(a) $k x e^{-2 x}$
(b) $k x e^{-4 x}$
(c) $k x e^{-r x}$

Integration Techniques
Improper Integrals
Mutahar Mehkri

Fundamental Theorem of Calculus

176 Practice Problems
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00:29
21st Century Astronomy

An empirical science is one that is based on
a. hypothesis.
b. calculus.
c. computer models.
d. observed data.

Motion of Astronomical Bodies
Eve Rafferty
00:41
Calculus: Early Transcendental Functions

Use the Fundamental Theorem of Calculus to find an antiderivative of $e^{-x^{2}}$

Integration
The Fundamental Theorem of Calculus
Ernest Castorena
01:09
Calculus: Early Transcendental Functions

Use Part I of the Fundamental Theorem to compute each integral exactly.
$$\int_{0}^{4} x(x-2) d x$$

Integration
The Fundamental Theorem of Calculus
Ernest Castorena

Indefinite Integrals

603 Practice Problems
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06:14
Calculus: Early Transcendental Functions

Find the volume of the solid formed by revolving the region bounded by $y=x \sqrt{\sin x}$ and $y=0(0 \leq x \leq \pi)$ about the $x$ -axis.

Integration Techniques
Integration by Parts
Anthony Ramos
04:12
Calculus: Early Transcendental Functions

Determine whether the integral converges or diverges. Find the value of the integral if it converges.
$$\int_{-\infty}^{\infty} \frac{1}{x^{2}} d x$$

Integration Techniques
Improper Integrals
Mutahar Mehkri
02:28
Calculus: Early Transcendental Functions

Evaluate the integral using integration by parts and substitution. (As we recommended in the text, "Try something!")
$$\int \sin (\ln x) d x$$

Integration Techniques
Integration by Parts
Darshan Maheshwari

Substitution Rule

307 Practice Problems
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01:36
Introductory and Intermediate Algebra for College Students

Solve the systems in Exercises $79-80$.
$$\left\{\begin{array}{l}
\log _{y} x=3 \\
\log _{y}(4 x)=5
\end{array}\right.$$

Conic Sections and Systems of Nonlinear Equations
Systems of Nonlinear Equations in Two Variables
Vishal Parmar
02:16
Introductory and Intermediate Algebra for College Students

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I think that the nonlinear system consisting of $x^{2}+y^{2}=36$ and $y=(x-2)^{2}-3$ is easier to solve graphically than by using the substitution method or the addition method.

Conic Sections and Systems of Nonlinear Equations
Systems of Nonlinear Equations in Two Variables
Vishal Parmar
02:23
Introductory and Intermediate Algebra for College Students

Explain how to solve a nonlinear system using the substitution method. Use $x^{2}+y^{2}=9$ and $2 x-y=3$ to illustrate your explanation.

Conic Sections and Systems of Nonlinear Equations
Systems of Nonlinear Equations in Two Variables
Vishal Parmar

Riemann Sums

142 Practice Problems
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04:16
Calculus: Early Transcendental Functions

Evaluate $\int_{0}^{2}\left[\tan ^{-1}(4-x)-\tan ^{-1} x\right] d x$ by rewriting it as a
double integral and switching the order of integration.

Multiple Integrals
Double Integrals
Uma Kumari
03:18
Calculus: Early Transcendental Functions

Compute the Riemann sum for the given function and region, a partition with $n$ equal-sized rectangles and the given evaluation rule.
$f(x, y)=x+2 y^{2}, 0 \leq x \leq 2,-1 \leq y \leq 1, n=4,$ evaluate
at midpoint

Multiple Integrals
Double Integrals
Uma Kumari
03:34
Calculus: Early Transcendental Functions

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums.
$$\int_{1}^{4} \frac{x^{2}}{x^{2}+4} d x$$

Integration
The Fundamental Theorem of Calculus
Adam Cornes

Net Change

71 Practice Problems
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05:12
Calculus for Scientists and Engineers: Early Transcendental

Air flow in the lungs A reasonable model (with different parameters for different people $)$ for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi V_{0}}{10} \sin \left(\frac{\pi t}{5}\right)$$, where $V(t)$ is the volume of air in the lungs at time $t \geq 0,$ measured in liters, $t$ is measured in seconds, and $V_{0}$ is the capacity of the lungs. The time $t=0$ corresponds to a time at which the lungs are full and exhalation begins.
a. Graph the flow rate function with $V_{0}=10 \mathrm{L}$
b. Find and graph the function $V$, assuming that $V(0)=V_{0}=10 \mathrm{L}$
c. What is the breathing rate in breaths/minute?

Applications of Integration
Velocity and Net Change
Kevin Luu
02:50
Calculus for Scientists and Engineers: Early Transcendental

Filling a tank A $200-\mathrm{L}$ cistern is empty when water begins flowing into it (at $t=0$ ) at a rate (in liters/minute) given by $Q^{\prime}(t)=3 \sqrt{t}.$
a. How much water flows into the cistern in 1 hour?
b. Find and graph the function that gives the amount of water in the tank at any time $t \geq 0.$
c. When will the tank be full?

Applications of Integration
Velocity and Net Change
Kevin Luu
11:12
Calculus for Scientists and Engineers: Early Transcendental

Where do they meet? Kelly started at noon $(t=0)$ riding a bike from Niwot to Berthoud, a distance of $20 \mathrm{km},$ with velocity $v(t)=15 /(t+1)^{2}$ (decreasing because of fatigue). Sandy started at noon $(t=0)$ riding a bike in the opposite direction from Berthoud to Niwot with velocity $u(t)=20 /(t+1)^{2}$ (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.
a. Make a graph of Kelly's distance from Niwot as a function of time.
b. Make a graph of Sandy's distance from Berthoud as a function of time.
c. How far has each person traveled when they meet? When do they meet?
d. More generally, if the riders' speeds are $v(t)=A /(t+1)^{2}$ and $u(t)=B /(t+1)^{2}$ and the distance between the towns is $D,$ what conditions on $A, B,$ and $D$ must be met to ensure that the riders will pass each other?
e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

Applications of Integration
Velocity and Net Change
Kevin Luu

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