What is Integrals: Definition and 1000 Discussions
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.
Homework Statement
\int_{C}(x+yz)dx + 2xdy + xyzdz
C goes from (1,0,1) to (2,3,1) and (2,3,1) to (2,5,2)
The Attempt at a Solution
For C going from (1,0,1) to (2,3,1)
x=1+t, y=3t, z=1; 0\leq t \leq 1
x'(t)=1, y'(t)=3, z'(t)=0
\int^{1}_{0}(1+t+3t)*1dt + 2(1+t)*3dt + 0...
Homework Statement
let c be the curve of intersection of the cone z= sqrt(x^2+y^2) and the plane 3z= y+4, taken once anticlockwise when viewed from above.
(i) evaluate
∫c (sinx - y)dx +(x+cosx)dy + (e^z + z)dz
(ii) let s be the surface of the cone z= sqrt(x^2+y^2) below the plane 3z=...
Homework Statement
I want to combine the 2 integrals:
\int_{a}^{b}(x-3)f(x)dx+\int_{-b}^{-a}(x-3)f(x)dx
Homework Equations
given:
f(x) is an even function
The Attempt at a Solution
swap the limits in the second integral:
\int_{a}^{b}(x-3)f(x)dx-\int_{-a}^{-b}(x-3)f(x)dx...
Homework Statement
int[sin(x)*cos(x)]dx
Homework Equations
U-substitution
The Attempt at a Solution
Alternative #1:
u=sin(x)
du=cos(x)dx
intdu = u^2/2 + K = sin^2(x)/2+K
Alternative #2
u=cos(x)
du=-sin(x)dx
-int[u]du = -u^2/2 + K = [U]-cos^2(x)/2+K
From...
I'm on q4 of this paper:
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2008/Paper63.pdf
For the first bit i said the coordinates had ranges t \in ( - \infty , \infty) , \quad \chi \in [0, 2 \pi)
Is that correct?
Anyway for the next bit we can take the equation g_{\mu \nu} u^\mu u^\nu...
Homework Statement
I have to calculate the following line integral
\int_{\gamma}y^{2}cos(xy^{2})dx + 2xycos(xy^{2})dy where \gamma is the path defined by the equations x(t) = t^{4} and y(t)=sin^{3}(\frac{t\pi}{2}) t between 0 and 1Homework Equations
Now I know that the formula for calculating...
If someone could link me to a tutorial on how to put in functions into a post, I would appreciate it, thanks. I am going to be putting in screen shots.
Homework Statement
http://img864.imageshack.us/img864/1517/scr1305133657.png"
http://img864.imageshack.us/img864/1517/scr1305133657.png...
Homework Statement
Evaluate the line integral yzdx+yzdy+ydz where C is the following semicircle The top half of y^2 + z^2 = 4 in the yz plane traveling from left to right.
Homework Equations
The Attempt at a Solution
What I tried, but I know it's not right, and I'm just not sure...
Homework Statement
solve integral x^3/(e^x-1) with limits from 0 to infinity
Homework Equations
The Attempt at a Solution
i tried using a rectangular contour,the boundaries of the contour pass through z=0 but the complex equivalent has pole at z=0. by Cauchy theorem the function...
Homework Statement
Find the Volume of the given solid
Bounded by the cylinders y^2+z^2=4 and x=2y, x=0,z=0 in the first octantHomework Equations
double integral over a region D with f(x,y) dAThe Attempt at a Solution
I graphed it in a xyz plane and got these intervals
D = {(x,y)|...
Homework Statement
Find surface integral of vector field F=<x,y,x+y> over the surface z=x^2+y^2 where x^2+y^2 less than 1. Use outward pointing normals
Homework Equations
The Attempt at a Solution
So I did the whole thing and got a zero which doesn't look right to me. My algebra...
Should Say Intervals.. I'm tired...
I am probably going wrong somewhere but I am running into problems with understanding this. My understanding of a 95% confidence interval is that in a sample of n the sample mean is 95% likely to be within 1.96 standard errors of the actual mean. I have a...
Homework Statement Given \mathbf{F} = \nabla f\; where \;f(x,y) = sin(x-2y)
Find a curve C that is not closed and satisfy the equation
\int_C \mathbf{F}\cdot dr = 0The Attempt at a Solution
\nabla f = \;<cos(x - 2y),-2cos(x-2y)>
So to satisfy the dot product being 0 (I am hoping I can do...
And why are the partitions not equal to one value? Why x1, x2, ... , xk, ... , xn-1, xn ?
And why |the norm| -> 0 ?
I was just curious if there is some specific logic behind it or if it is just there to discuss things in general.
Thanks a lot.
P.S.: Norm is the partition having the...
Homework Statement Prove the Mean Value Theorem for Integrals
Proof
Let f(x) be defined on [a,b]
Let M be the max of f(x) and m be the min of f(x)
Then
m \leq f(x) \leq M
\int_{a}^{b}m \;dx\leq \int_{a}^{b} f(x)\;dx \leq \int_{a}^{b} M\;dx
m(b-a) \;dx\leq \int_{a}^{b} f(x)\;dx \leq...
Homework Statement
Evaluate this integral using trigonometric substitution.
\int_{0}^{2} \frac{x^3}{\sqrt{4-x^2}}dx
Now I can do this the "textbook memorization" method like every calculus student does, but I want to go ahead an analyze this further. But I will show you the...
In various quantum chemistry books and course booklets I came across spin wave functions (usually referred to as alpha and beta for spin and up and down, respectively) that depend on a so called spin-variable. They are usually used to construct slater determinants. An example of this is Modern...
Homework Statement
find volume of solid bounded by z=x, y=x, x+y=2 and z=0
The Attempt at a Solution
first need to find domain.
for x bounds, when y=0, x=0, when y = x, x+x=2 so x=1 therefore 0 < x < 1
for y bounds, x < y < 2-x
now I am trying to work out what i integrate...
We are given a vector field:
F=\frac{-y}{x^2+y^2} , \frac{x}{x^2+y^2}
Then asked if F is conservative on R2 \ (0,0). I just solved the partial derivatives of each part of the vector field and they did indeed equal each other, but I don't under stand what the "\(0,0)" part means.
We are then...
Hi i need to create a MATlab m file solving the following function for 0 to 90 degrees of \theta_0 and for any function F(\theta^').
[PLAIN]http://rogercortesi.com/eqn/tempimagedir/eqn9903.png
[edit] dx should be d\theta^', sorry about that.
I managed to do it in MATlab using symbolic maths...
Homework Statement
The question is to use MATLAB to evaluate a triple integral in spherical coordinates to find the mass density of the solid inside the cone z = (3x^2 + 3y^2)^.5 and below z = 5 where the mass density at (x,y,z) is equal to the z coordinate of the point.
Homework...
I already posted about this but I redid the problem and got another answer:
1. Homework Statement
The volume of an air mattress is 15 cubic feet. Air escapes at a rate of r(t)=0.25e^(−0.05t), where r is in cubic feet per second. Assuming the mattress is still completely inflated when the...
Homework Statement
Evaluate the integral with respect to x from 0 to infinity when the integrand is x^2/(1+x^6), using complex integration techniques.
Homework Equations
The Attempt at a Solution
I have no idea where to start. Please help!
Hello!
I know that the theory of complex analysis is useful to compute integrals of real valued functions. I am a Physics student and I followed a Complex Analysis course but we did not have time to cover this up.
I am looking for a textbook that takes a practical approach to this subject. I...
Hello dear colleagues!
Yesterday i was trying to proof the surface area of a sphere formula, then i got some problems. I know that something is seriously wrong in this concept, but i can't tell what exactly is wrong. Could you guys help me please?
I just thougt about a hollow sphere, then we...
Homework Statement
It's a physics problem, where i have to evaluate the root-mean-square radius defined by the expression below. (First for a constant \rho, then for a "(r)dependent" \rho).Homework Equations
(\int{_0}{^\infty} \rho r^4 dr / \int{_0}{^\infty} \rho r^2 dr) ^(1/2)The Attempt at a...
Homework Statement
Use a triple integral to calculate the volume of the solid enclosed by the sphere
x^2 + y^2 + z^2=4a^2 and the planes z=0 and z=a
Homework Equations
Transform to spherical coordinates (including the Jacobian)
The Attempt at a Solution
I'm stuck, as the radius...
Homework Statement
\int\int\int^{}_{B} ye^(-xy) dV where B is the box determined by 0 \leq x \leq 4, 0 \leq y \leq 1, 0 \leq z \leq 5.Homework Equations
The Attempt at a Solution
\int^{4}_{0}\int^{1}_{0}\int^{5}_{0} ye^(-xy) dzdydx
Integrating the first time I get
zye-xy
Plugging in 5 and 0 I...
Suppose that the bounded function f:[a,b]-->R has the property that for each rational number x in the interval [a,b], f(x)=o for all x in [a,b]. Prove that
the lower integral of f from a to b is less than or equal to zero which is less than or equal to the upper integral of f from a to b...
I am trying to find the region between a surface z= x+4y and the region D in the x-y plane, where the region is the triangle with verticies (1,1) (2,3) (0,0).. However I am not sure how to come up with the double integral?
Homework Statement
Find the centroid x,y,z of the region R cut out of the region 0<=z<=5sqrt(x2+y2) by the cylinder x2+y2=2x.
Homework Equations
x2+y2 = r2
x= rcosθ
y= rsinθ
The Attempt at a Solution
Centroid x being Mx/m I'm guessing
I've been working on this problem...
“Non-integrable” multiple integrals for Mathematica
Dear all,
I have been trying to crack one problem in Mathematica, but I keep getting a wrong answer probably because I have something either fundamentally wrong analytically or code wise. OK, here is the problem.
Suppose we have to...
\int^{1}_{0}\int^{0}_{-x} \frac{ysin(pi*y^2)}{1+y} dydx
Not exactly sue how to start this. I know that I need to integrate with respect to y first then use that solution and integrate again with respect to x however I do not believe integrating the initial problem is possible. Is there another...
#1
Homework Statement
\int r^4 ( ln (r) ) drHomework Equations
Infinity algebra and Calc related formulae..The Attempt at a Solution
Not sure even where to start here.. I'm thinking a u-substitution, letting u = r^3 so that I can deal with the two left over r's, but I don't think that it would...
Homework Statement
Find the volume of the cone bounded below by z=2root(x2+y2) and above by x2 + y2 + z2 = 1
Homework Equations
The Attempt at a Solution
Ok I have the solution, I just don't understand how to get it!
So I know I have to change into spherical coordinates but...
Homework Statement
Evaluate the double integral ∫∫D xy dA where D is the triangular region with vertices (0,0) (6,0) (0,1).
Homework Equations
The Attempt at a Solution
0 <= x <= -\frac{1}{6}x+1
0 <= x <= 6
the first integral would be the integral from 0 to -1/6x+1 of xy with...
find the integral int(1/z)dz along r for the curve:
square with corners 1+i, -1+i, -1-i, 1-i
traversed clockwise and anti-clockwise
Homework Equations
i know that clockwise will be the -(int) of the anticlockwise
The Attempt at a Solution
the first line = (1-2t)+i it's derivative...
I don't speak English very well, so it's very hard to me to explain my attemps to solve this problem, and I'm still learning to use latex, so it's so slow to me. I can scan my attemps if you want to see them.
Homework Statement I_n = \int_{0}^{\infty} x^{2n-1}/(x^2+1)^{n+3} \dx, n \geq 1
I...
Homework Statement
Im getting very confused with working out how to integrate the following double integral with an absolute value:
\int^{2}_{-2}\int^{2}_{-2}\left|x^{2}+y^{2}-1\right|dxdy
Homework Equations
The Attempt at a Solution
I know you have to split it down into where it...
Calculus: Coordinate Changes, Jacobian, Double Integrals??
Homework Statement
Show that T(u,v) = (u2 - v2, 2uv)
maps to the triangle = { (u,v) | 0 ≤ v ≤ u ≤ 3 } to the domain D,
bounded by x=0, y=0, and y2 = 324 - 36x.
Use T to calculate ∬sqrt(x2+y2) dxdy on the region D...
Homework Statement
Solve for the volume above the xy-plane and below the paraboloid z=1-x2/a2-y2/b2
I have gotten an answer that is close to the correct one, but I can't figure out where I am wrong.
Homework Equations
Solution: Volume is = ab\pi/2
The Attempt at a Solution...
To find the probability of a particle being at position x we use
<\Psi|\Psi> where the complex conjugate ensures that the answer is real. This means that we're looking at the square of the wave function to determine the probability of finding the particle.
Now to determine the probability...
Homework Statement
Find a transform T that maps the unit square in the u-v plane to a quadrilateral with corners (1,2), (3,3), (4,2) and (2,1) to the x-y plane.
Homework Equations
The Attempt at a Solution
I've been able to create the proper region in the x-y plane when I have the transform...
Homework Statement
Convert the Riemann's Sum to an integral:
(1/50) * [(sqrt(1/50)) + (sqrt(2/50)) + (sqrt(3/50)) ... + (sqrt(50/50))]
Homework Equations
The Attempt at a Solution
(1/50) times Integral (upper limit 1 and lower limit 0) of sqrt(x) dx
Is my solution correct?
Homework Statement
This is just something I've been wondering, but since derivatives have the formula:
dy/dx = lim h-> 0 of (f(x+h) - f(x)/h)
And that formula can prove a lot of derivatives.
Does a similar formula exist that can prove integrals?
Homework Equations
The Attempt...
Homework Statement
A voltage of 60cos(4 \pi t) V appears across the terminals of a 3mF (milli-farad) capacitor. (which is equal to .003 F [farad]).
Calculate the current through the capacitor and the energy stored in it from t=0 to t=.125 s
Homework Equations
Current through an ideal...
Homework Statement
Well the problem asks for flux of B threw the loop
its a square loop with length a sides and its a length S above a wire
Homework Equations
The Attempt at a Solution
so the integral i am getting confused about is
\int\muI/(2pi s) da da is area
now moving...
Hi, I am currently in Calculus 2 at my local college and I am having trouble wrapping my head around Improper Integrals. The question below I have been working on for awhile and I think i have an answer but was wondering if anyone could confirm if I was thinking about this question the right...
This is probably an easy question, but my math is not good enough to answer it.
For Gaussian integrals:
\frac{\int \Pi_i [dx_i] x_k x_l e^{-\frac{x_i A_{ij} x_j}{2}}} {\int \Pi_i [dx_i] e^{-\frac{x_i A_{ij} x_j}{2}}}=A^{-1}_{kl}
As far as I understand it, in QFT, Aij is a local operator. So...
Homework Statement
prove that
2√2 <= ∫(from 0 to 1) (√x+8) dx <= 3
Homework Equations
The Attempt at a Solution
well...my only idea on how to solve this would be to evaluate the middle term, but my prof says it's not allowed. Do I just assign functions to the left and right...