integrals Definition and Topics - 43 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.

View More On
  1. Adesh

    I Why integrals took 2000 years to come up in a rigorous manner?

    Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer...
  2. Adesh

    How to prove that ##M_i =x_i## in this upper Darboux sum problem?

    We're given a function which is defined as : $$ f:[0,1] \mapsto \mathbb R\\ f(x)= \begin{cases} x& \text{if x is rational} \\ 0 & \text{if x is irrational} \\ \end{cases} $$ Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 ...
  3. fresh_42

    Applied Integrals, Series and Products

    I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.
  4. Adesh

    I Understanding the ##\epsilon## definition of this integral

    Integrals are defined with the help of upper and lower sums, and more number of points in a partition of a given interval (on which we are integrating) ensure a lower upper sum and a higher lower sum. Keeping in mind these two things, I find the following definition easy to digest A function...
  5. O

    I Is my derivation correct?

    My work is in the following pdf file:
  6. Adesh

    Why does dividing by ##\sin^2 x## solve the integral?

    If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...
  7. R

    Integral of Acceleration with respect to time

    Homework Statement Acceleration is defined as the second derivative of position with respect to time: a = d2x/dt2. Integrate this equation with respect to time to show that position can be expressed as x(t) = 0.5at2+v0t+x0, where v0 and x0 are the initial position and velocity (i.e., the...
  8. Saurabh

    Hairy trig integral

    <Moderator's note: Moved from a technical forum and thus no template.> where a, b, c, d and n, all are positive integers. Find the value of 'c'. ------------------------------- I don't really have a good approach for this one. I just made a substitution u = sinx + cosx I couldn't clear up...
  9. GaussianSurface

    Calculating distance from speed

    Homework Statement The speed of a runner increased during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she travelled during these three seconds. It follows the image's square. Homework...
  10. M

    Finding population density with double integrals

    Homework Statement A city surrounds a bay as shown in Figure 1. The population density of the city (in thousands of people per square km) is f(r, θ), where r and θ are polar coordinates and distances are in km. (a) Set up an iterated integral in polar coordinates to find the total population...
  11. M

    Proving the equation for the height of a cylinder

    Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...
  12. W

    Contour Integrals: Working Check

    Homework Statement Hi all, could someone help me run through my work for these 2 integrals and see if I'm in the right direction? I'm feeling rather unsure of my work. 1) Evaluate ##\oint _\Gamma Z^*dz## along an anticlockwise circle of radius R centered at z = 0 2) Calculate the contour...
  13. W

    Solving an Integral

    Homework Statement Solve from x = 0 to x = ∞, ∫xe-axcos(x)dx Homework Equations The Attempt at a Solution I have a solution for the integral ∫e-axcos(x)dx at the same limits, and I feel that the result might be of use, but have no idea how to manipulate the integral above such that I can use...
  14. D

    Determine the truth of the following statements

    Homework Statement ##f(x) = \begin{cases} -\frac{1}{1+x^2}, & x \in (-\infty,1) \\ x, & x \in [1,5]\setminus {3} \\ 100, & x=3 \\ \log_{1/2} {(x-5)} , & x \in (5, +\infty) \end{cases}## For a given function determine the truth of the folowing statements and give a brief explanation: a) Function...
  15. Kumaken

    Application of integrals to find moment Inertia

    I need to make a project that integrates physics with math, involving the use of integrals to find moment inertia of areas. The theory could be read here : I need to make an object that applies the theory above. Can anybody...
  16. doktorwho

    Best resources for learning Integrals

    What would tou suggest as the best resource for learning integrals? I need preferably some practical books videos or youtube channels that deal with application and problems rather than theory. Any thoughts? Thanks
  17. doktorwho

    Integration by substitution question

    Homework Statement Question: To solve the integral ##\int \frac{1}{\sqrt{x^2-4}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##? Explain Homework Equations 3. The Attempt at a Solution [/B] This is my reasoning, the function ##\operatorname...
  18. cg78ithaca

    A Inverse Laplace transform of a piecewise defined function

    I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...
  19. cg78ithaca

    A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

    This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
  20. D

    A Fourier Integral help

    Where , rho 1 and rho 2 are two dimensional position vectors and K is a two dimensional vector in the Fourier domain. I encountered the above Eq. (27) in an article and the author claimed that after integration the right hand side gives the following result: I tried to solve this integral but...
  21. bwest121

    How do I calculate this integral?

    Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...
  22. O

    Integral sqrt(x)*e^-x

    Homework Statement Evaluate the following integral: ∫0∞ √(x)* e-x dx Homework Equations ∫0∞ e-x2 dx = (√π)/2 The Attempt at a Solution So far this is what I've done: u = x1/2 du = 1/2 x-1/2 2 ∫ e-u2 u2 du Now, I'm not really sure what to do? Or if what I've done so far is leading me down...
  23. F

    I Definitions of the Riemann integral

    In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...
  24. egio

    I Difference between integral and antiderivative?

    Hello, I still don't really understand what an antiderivative is, besides its ability to "undo" derivatives, its relation to integrals, and what the difference between the two even is. It would also be great to know how to visualize an antiderivative. I've tried looking further into the...
  25. heartshapedbox

    Finding future x position with changing velocity

    Homework Statement At what time in the future will the x-component of the masses position be at 0m? Homework Equations x=x,initial+(integral0,t)(v,xcomponent)(dt) The Attempt at a Solution The solution is 3.1 seconds... not really sure where to start :(
  26. M

    Volume Integral

    Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the...
  27. E

    Integrals to Solve Area and Center of Mass of a Cut Circle

    Homework Statement I am after finding the centroid of the remaining area (hatched) when a circle is cut by a line. I made a diagram in CAD that demonstrates the problem. The idea is that, starting from the bottom of the circle, a cut is taken leaving a remaining shape whose area and...
  28. T

    Surface area of a sphere with calculus and integrals

    Homework Statement How do I find the surface area of a sphere (r=15) with integrals. Homework Equations Surface area for cylinder and sphere A=4*pi*r2. The Attempt at a Solution I draw the graph for y=f(x)=√(152-x2). A circle for for positive y values which I rotate. I will create infinite...
  29. N

    Upper and lower bound Riemann sums

    Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...
  30. N

    Compute upper and lower integral

    Homework Statement Let f(x) = x^2 and let P = { -5/2, -2, -3/2, -1, -1/2, 0, 1/2 } Then the problem asks me to compute Lf (P) and Uf (P). Lf (P) = Uf (P) = The Attempt at a Solution Please explain how to solve. I thought that L[f] meant to calculate the lower bound with respect to f(x)...