In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.
1) Find the radius of curvature at any point of the cycloid x = a(\theta + sin\theta)y = a(1- cos\theta).
2) Find the radius of curvature at the point (3a/2 , 3a/2) for the curve x3 + y3 = 3axy
Homework Statement
For an analytic function, f(x+iy)=u(x,y)+iv(x,y),u is given by u=(3x^2) -(3y^2). The expression for v, considering K to be a constant is?
Homework Equations
δu/δx=δv/δy
δu/δy=-δv/δx
[/B]
The Attempt at a Solution
My attempt :-
u=(3x^2) -(3y^2)
δu/δx=6x &...
Homework Statement
i want to find limit value using riemann sum
\lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br>
question : <br>
\lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br>
Homework EquationsThe Attempt at a...
I have learned an adequate amount of calculus including implicit, parametric differentiation as well as Upton second order differential equations in high school math course. It was really abstract and we were taught only how to solve mathematical problems. Now, I need to model those problems in...
Homework Statement
the moment of inertia of A cylinder of height 2h radius (a) and uniform mass density ρ about a line x=y=z using multiple integration.
Homework Equations
I=ρ∫s^2*dV where the integral is over the volume V of cylinder and s is the perpendicular distance to the axis of...
Homework Statement
Can this function be integrated analytically?
##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
\sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),##
where ##a##, ##b## and ##L## are some real positive...
Homework Statement
I have a 2D integral that contains a delta function:
##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##,
where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
I have a short doubt: Let f(x) be a fuction that can't be integrated in an analytical way . Is anything wrong if I expand it in a Taylor' series around a point and use this expansion to get the value of the definite integral of the function around that point? Suppose that the interval between...
Is it possible to integrate the following function analytically?
##\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,##
where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded...
I am trying to numerically integrate the following complicated expression:
$$\frac{-2\exp{\frac{-4m(u^2+v^2+vw+w^2+u(v+w))}{\hbar^2\beta}-\frac{\hbar\beta(16\epsilon^2-8m\epsilon(-uv+uw+vw+w^2-4(u+w)\xi...
Homework Statement
http://imgur.com/a/k7fwG
Find the vector magnetic potential at point P1.
Homework Equations
Vector magnetic potential given by:
$$
d \bar{A} = \frac{\mu I d\bar{l'}}{4 \pi | \bar{r} - \bar{r'} | }
$$
The Attempt at a Solution
I split up the problem in 3 parts,
first...
Homework Statement
A homopolar generator consists of a metal disc of radius ##a## and a central axle which has radius ##a/4##. The disc has resistivity ##\rho## and thickness ##t##. It is rotated in a uniform magnetic field ##B## about an axis through the centre, which is parallel to ##B## and...
Homework Statement
Calculate the indefinite integral of the function ## \int\frac{3x^3}{\sqrt{1-x^2}}##
my book gives the answer ##-(2+x^2)\sqrt{1-x^2}+C##
Homework EquationsThe Attempt at a Solution
So I started trying to calculate this indefinite integral by using a substitution...
Hi All,
I am wondering if the function below is Integrable:
$$\frac{\exp{(-\frac{1}{2}(u-2)^2-2u^2)}}{u-2}$$
When I work it out on computer, the integral is finite from -Inf to Inf. But clearly it has a pole at u=2. Is this pole integrable? If yes, what kind of coordinate transform is...
Note: I didn't really know where to put this. It isn't a specific problem, but I've been asked by my physics teacher, who decided to give me and a few others an individual physics course of sorts, to find the means of solving similar problems. It's the first problem he assigned us, since we're...
Homework Statement
I'm trying to find the value of the integration:
Homework Equations
Mod note: Edited the TeX to render properly at this site.
Here is the integration written using MAketex:
##\frac{.5}{\sqrt{\pi}}\int_0^{\infty}\exp(-z(1+1.5/v))z^{L-.5} \frac{1}{(\frac{z}{v}+1)^n...
Homework Statement
The problem asks to find g'(2), g''(2), and g'''(4).
Homework Equations and attempt at solution[/B]
The derivative of g(x) is just the function f(x). So g'(2) = f(2) = -2.
I'm not sure how to find g''(2) and g'''(4).
I understand that g''(2) is f'(2), but how do I find...
HELP I can't find the surface of revolution! By donuts I mean a circle that doesn't touch the axes (tore in french)
y^2+(x-4)^2=2^2 is my function ( y^2+x^2=r^2) and the axe of rotation is y
so y= sqrt(r^2-x^2)
the formula I know :
2* pi (Integral from a to b (F(x)*sqrt( 1+ (f``(x))^2))...
Suppose ##\intop_{-\infty}^{+\infty}(f(x))^{2}dx=1##, and ##a=\intop_{-\infty}^{+\infty}(\frac{df(x)}{dx})^{2}dx##, does a maximum value of ##a## exist? If it exists, what's the corresponding ##f(x)##?
Homework Statement
The question states Use integral calculus to find the euation of the quartic that has (1,23) and (3, 15) and a y-intercept of 24.
Homework Equations
The previous part of the question was A quartic has stationary points of inflection at x=1 and x=3. Explain why...
I've been trying to prove a couple of properties of integrals using the Riemann sum definition: $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned (such that ##a=x_{1}<x_{2}<\cdots <x_{i-1}<x_{i}<\cdots...
I am trying to calculate the magnetic field generated by an ideal toroidal solenoid by using the integral of the Biot-Savart law. I do not intend to use Ampère's circuital law.
Let ##I## be the intensity of the current flowing in each of the ##N## loops of the solenoid, which I will consider an...
My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density ##n## (i.e. there are ##n## loops per length unit), a current of intensity ##I## flows, has the same direction as the loops' moment of magnetic dipole and magnitude ##\mu_0...
Homework Statement
In a cubical volume, 1.05 m on a side, the electric field is given by the formula below, where E0 = 1.25 N/C and a = 1.05 m.
= E0(1 + z/a) i + E0(z/a) j
The cube has its sides parallel to the coordinate axes, see the figure. Determine the net charge within the cube...
NOTE: Other threads suggest solving it with Gauss' Law. I'd like to see an approach through direct integration, no full followthrough necessary..
1. Homework Statement
Consider a sphere with a uniform distribution of charge ρ (ro). Inside the sphere is a cavity (spherical). Calculate the...
Homework Statement
Q=-1*K(T)*(H*W)*(dT/dx)+((I^2)(p)(dx)/(H*W))
K(T)=(197.29-.06333333(T+273))
H=0.01905
W=0.06604
I=700
p=10*10^-6
Q=some constant
Please separate and differentiate to solve for Q using variables of T and x.
Boundaries:
T: Upper=T1 (constant)
Lower=T0 (constant)
x: Upper=L...
Homework Statement
D1=6cm D2=2cm ( i have to prove that 2nd moment of area (J) of a circulal plate abouts its polax axis(zz) is equal to piD^4/32 )
Homework EquationsThe Attempt at a Solution
J=pi6^4/32 - pi2^4/32 = 125.66
J=r1^2x2pirdr
J=Integral 2pir^3d2 = 2pi integral(high 3 , low 1)...
Homework Statement
Solve the following equation.
Homework Equations
( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0
The Attempt at a Solution
M = ( 3xy4 + 2xy )
N = ( 2x3y3 - x2 )
∂M/∂y = 12x2y3 + 2x
∂N/∂x = 6x2y3 - 2x
Then this equation looks like that the integrating factor is (xM-yN).
IF =...
Homework Statement
The total area between a straight line and the parabola is revolved around the y-axis. What is the volume of revolution?
According to the book, the answer is ; My answer comes out to be
Homework Equations
The Attempt at a Solution
1. Rewrite the second equation in...
I know that \sqrt{f(x)^2} = |f(x)| However...
I've just noticed that integrals of expressions like this are usually assumed to be equal to the integral of f(x) without the absolute value. I'd like to know how that's possible.
Is weird for me to consider those expressions; specially because of...
The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law essentially uses the fact that, if ##\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3##, ##\boldsymbol{J}\in C_c^2(\mathbb{R}^3)##, is a compactly supported twice continuously differentiable field...
Find the volume of the solid generated by revolving the region bounded by y = 4x-x^2 and y = 2x about the y-axis. About the line x = 2.
Since this is rotated about the y-axis, I know I have to manipulate the equation so I can get to x = something. The problem is I cannot change the equation y =...
Hey! I want to do a double integral calculation of this problem##∫∫ xy/(xy^2 +1)^2##
over the region bounded by 2 ≤ x ≤ 3 and 2*sqrt(1+x) ≤ y ≤ 2*sqrt(2+4x)
on MATLAB and i have tried the following syntax:
clc
clear all
fun=@(x,y) x*y./((x*y.^2+1).^2);
ymax=@(x) 2*sqrt(2+4*x)...
This is the key step to transform from position space Schrodinger equation to its counterpart in momentum space.
How is the first equation transformed into 3.21?
To be more specific, how to integral Laplacian term by parts?
Homework Statement
Verify the divergence theorem for the function
V = xy i − y^2 j + z k
and the surface enclosed by the three parts
(i) z = 0, s < 1, s^2 = x^2 + y^2,
(ii) s = 1, 0 ≤ z ≤ 1 and
(iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1.
Homework Equations
[/B]...
Homework Statement
Evaluate the integral to find the area.
Homework Equations
The Attempt at a Solution[/B]
gifs upload
So I know how to find an anti-derivative for the most part. Here it's tricky because my equation has an exponent AKA square root. I tried to use the chain rule with...
Homework Statement
The probability that a particular computer chip fails after a hours of operation is given by
0.00005∫e^(-0.00005t)dt on the interval [a, ∝]
i. find the probability that the computer chip fails after 15,000 hours of operation
ii. of the chips that are still operating after...
Homework Statement
Find the circulation (line integral) of y2dx+x2dy for the boundary of a triangular region contained within x+y=1, x=0, and y=0.
Homework Equations
Green's theorem
The Attempt at a Solution
I think I actually already got the solution; I used the Green's theorem to get the...
Good afternoon,
i was just wondering if this equation is possibly solvable where I(z) is a function of z. The equation is:
I(z)=cosh(1/2 ∫I(z)dz)
I know it looks stupid but is it possible? How would you approach this problem?
Thank you.
Hello,
Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...
Hello,
Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...
Homework Statement
y'' + 3y' + 2y = r(t),
r(t) = u(t - 1) - u(t - 2),
y(0) = y'(0) = 0.
I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is?
Homework Equations...
Homework Statement
Evaluate \int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx via trigonometric substitution.
You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution.
Homework EquationsThe Attempt at a Solution
Letting x=sinθ...
Hi,
I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second...
The name of lectures of integral calculus written by Johann or Jeans Bernoulli (he is called by both names as far as I know) might be " lecciones mathematicæ de calculo integral". I searched for one day for the english translation, I couldn't even find the english title of his book on integral...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
1) ∫ dx/((x^(2))(x^(2)+4)^(1/2))2) I'm really stumped on this integral. I've tried several different methods of integration, but I kept getting stuck.3) The problem doesn't look like it needs trig...
Say we have a function of the form f(x)=\bigg\lbrace\begin{matrix}c \qquad\quad\text{x=0}\\ 0\quad\text{elsewhere}\end{matrix} If we then integrate this over all space i.e. \int_{-\infty}^{\infty}f(x)dx Why does the result equal zero?
Kindly can someone tell me how can i integrate and plot the following formula as a function of time. while theta is changing from 1 to 20.
also attached .docx file for convenience. Any kind of help or advice will be highly appreciable.