What is Definite integral: Definition and 390 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.

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  1. S

    Use fundamental theorem of calculus to compute definite integral

    Homework Statement The problem and my (incorrect) work are typed and attached as TheProblemAndMyWorkTypedUp.jpg. Homework Equations Integral from a to b of f(t) = F(b) – F(a) The Attempt at a Solution As mentioned above, my work is attached as TheProblemAndMyWorkTypedUp.jpg. (The (2 –...
  2. anemone

    MHB How Challenging Is This Definite Integral with a Tangent and Pi Power?

    Evaluate \int_0^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{\pi e}}.
  3. anemone

    MHB Evaluate Definite Integral Challenge

    Evaluate \int_0^{\pi} \frac{\cos 4x-\cos 4 \alpha}{\cos x-\cos \alpha} dx
  4. paulmdrdo1

    MHB Solve Definite Integral: Invalid Answer Error

    i tried to solve this definite integral but i keep on getting an invalid answer. please check my error. $\displaystyle \int_{-3}^{-2}\frac{y+2}{y^2+4y}dy$ $\displaystyle u=y^2+4y$ $\displaystyle du=2y+4dy$ $\displaystyle dy=\frac{du}{2y+4}$ $\displaystyle \frac{1}{2}\int\frac{y+2}{u}\times...
  5. MarkFL

    MHB Maya's question at Yahoo Answers regarding a Riemann sum and definite integral

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  6. anemone

    MHB How Can You Effectively Evaluate a Challenging Definite Integral?

    Evaluate \int_2^4 \frac{\sqrt{\ln (9-x)}\,dx}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}
  7. MarkFL

    MHB Do's question at Yahoo Answers regarding differentiating a definite integral

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  8. Y

    Evaluating Definite Integrals: A Comparison and Explanation

    I want to proof \int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta=\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta (1) Let ##\theta=-\theta\;\Rightarrow\; d\theta=-d\theta,\;\cos(-\theta)=\cos\theta,\;\cos[m(-\theta)]=\cos(m\theta)## \int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta)...
  9. anemone

    MHB Find The Exact Value of A Definite Integral

    Find the exact value of the following definite integral: \int_0^2 (3x^2-3x+1)\cos (x^3-3x^2+4x-2)\,dx
  10. MarkFL

    MHB Stardock's question at Yahoo Answers regarding minimizing a definite integral

    Here is the question: I have posted a link there to this topic so the OP can find my work.
  11. B

    MHB Graphing definite integral functions

    Hi, how are you? I came across some exercises that really puzzled me. They ask me to graph the following functions: a) \int_0^x\sqrt{|tan(w)|} dw b)\int_0^\sqrt{x} e^{t^2}I imagine I'll have to use derivative techniques as I would when graphing a "normal" function, but those integral signs...
  12. J

    Evaluating a definite integral when a condition is given

    Homework Statement Given that x^{2}f(x)+f(\frac{1}{x})=0, then evaluate \int^{1.5}_{0.6}f(x)dx Homework Equations The Attempt at a Solution tried to replace f(x) using the provided equation...didn't help
  13. Saitama

    Simplifying Definite Integrals with Quotient Rule

    Homework Statement If ##\displaystyle P=\int_0^{\pi} \frac{\cos x}{(x+4)^2}dx## and ##\displaystyle I=\int_0^{\pi/2} \frac{\sin (2x)}{2x+4}dx##, then the value of ##P+2I-\frac{1}{\pi+4}## is equal toHomework Equations The Attempt at a Solution By substituting 2x=t i.e 2dx=dt, and replacing t...
  14. B

    MHB Definite integral involving the natural log function

    I figured I would just add this new problem over here, rather than starting a new thread. Im looking to solve integration leading to arctan or arcsin results. \int_{1}^{e}\frac{3dx}{x(1+\ln(x)^2}) Looking at this, it feels like this has an arctan in the result, but I would have to multiply...
  15. F

    Evaluate the definite integral

    1. \int^{a}_{0} x\sqrt{x^{2}+a^{2}} a > 0 2. u = x^{2} + a^{2}, du = 2x 3. \frac{1}{2}\int^{a}_{0}\frac{2u^{3/2}}{3} = \frac{1}{3}\int ^{a}_{0}(x^{2}+a^{2})^{3/2} How do I solve this? Any hints?
  16. 1

    Evaluate the definite integral:

    Homework Statement Evaluate the definite integral ∫(x101-√(9-x^2))dx from -3 to 3. Hint: This problem can be done without anti-differentiation.Homework Equations The Attempt at a Solution I am stuck. I tried to do it with with anti-differentiation and it didn't work/very...
  17. Saitama

    Function involving definite integral

    Homework Statement Let g be a continuous function on R that satisfies ##\displaystyle g(x)+2\int_{0}^{\pi/2} \sin x \cos t g(t)dt=\sin x##, then ##g'\left(\frac{\pi}{3}\right)## is equal to A)1/2 B)1/√2 C)1/4 D)none of these Homework Equations The Attempt at a Solution...
  18. R

    Help needed regarding proof of definite integral problem

    Homework Statement f(x) is a bounded function and integrable on [a,b] . a, b are real constants. We have to prove that i) An = a∫b f(x)cos(nx) dx → 0 when n→∞ ii)Bn = a∫b f(x)sin(nx) dx → 0 when n→∞ Homework Equations Parseval's formula : For uniform convergence of f(x) with its...
  19. P

    Definite Integral of Absolute Value Function (Calc I)

    Homework Statement $$\int_{0}^{\ 2\pi} \ |e^{sin(x)}cos(x)| \, dx$$ I know that it simplifies to $$ 2e- \frac{2}{e} ≈ 4.7 $$ I'm not sure how to approach this problem. Do I just break the integral up into the domains where it's positive and negative and integrate each component...
  20. Saitama

    Integrating Trigonometric Functions with Exponential Factors

    Homework Statement \int_{0}^{2\pi} \frac{dx}{1+e^{\sin x}}Homework Equations The Attempt at a Solution Let ##I=\int_{0}^{2\pi} \frac{dx}{1+e^{\sin x}}## Since ##\int_{a}^{b}f(x) dx=\int_{a}^{b} f(a+b-x)dx## Hence, I=\int_{0}^{2\pi} \frac{dx}{1+e^{-\sin x}}=\int_{0}^{2\pi} \frac{e^{\sin...
  21. J

    Can the Integral of a Complex Gaussian Function Be Expressed in Closed Form?

    Is there a closed form expression for the following definite integral? \int_{-∞}^{∞} exp(\frac{-|z|^2}{2{\sigma}^2}-\alpha |\mu + z|)dz where z is complex, and \alpha, \sigma, \mu are real constants. I couldn't obtain an expression similar to Gaussian integral, so I couldn't take the...
  22. MarkFL

    MHB Susie's question at Yahoo Answers regarding minimizing a definite integral

    Here is the question: Here is a link to the question: Calculus question, please, please answer.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  23. C

    Definite integral with variable limits of a multivariable function.

    I have the following integral: \int_0^{f(x,y)}{f' \sin(y-f')df'} Now suppose that f(x,y) = x*y, my question is how do I write the integral in terms of x and y only? Can I do something like this? Since df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy we can obtain...
  24. Saitama

    How can I evaluate this definite integral with conflicting results?

    Homework Statement I=\int_0^{\pi} \frac{xdx}{9\cos^2 x+\sin^2 x} Homework Equations The Attempt at a Solution The given integral can be written as I=\int_0^{\pi} \frac{(\pi-x)dx}{9\cos^2 x+\sin^2 x} The denominator remains unchanged because ##\cos^2(\pi-x)=-\cos x## and square...
  25. P

    Is this Tricky Definite Integral Solvable?

    1. The problem, the whole problem, and nothing but the problem \int_0^\pi \frac{x \cdot sin(x)}{1+cos^2(x)} \, dx Homework Equations Integration by parts trig substitution The Attempt at a Solution My first idea was to break up the integral by letting u=x and dv=sin(x)/(1+cos^2 x). I will...
  26. 5

    Process for finding a definite integral

    Homework Statement given that ∫01 xndx = 1/ n+1 for integers n \geq 0 calculate these integrals Homework Equations ∫01 (1+x+x2) dx The Attempt at a Solution I have found the definite integral of ∫01 (1+x+x2) dx to = 1.8333... (ie. 11/6) but all I did was use my TI-84 to find...
  27. MarkFL

    MHB Integration by Parts: Calculus Integral Help?

    Here is the question: Here is a link to the question: Calculus integral help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  28. T

    Definite integral: why is this wrong?

    According to wolfram alpha, the correct answer is -3/4.
  29. Saitama

    Solving Definite Integral Homework: f(x)=4x^4-24x^3+31x^2+6x-8

    Homework Statement Consider ##f(x)=4x^4-24x^3+31x^2+6x-8## be a polynomial function and ##\alpha, \beta, \gamma, \delta## are the roots of the equation ##f(x)=0##, where ##\alpha < \beta < \gamma < \delta##. Let sum of two roots of the equation f(x)=0 vanishes. Then the value of...
  30. MarkFL

    MHB Mohammad nabeel's question at Yahoo Answers regarding a definite integral

    Here is the original question: Here is a link to the question: Ntegral of sin^3(6x)cos^4(3x) dx ? - Yahoo! Answers I have posted a link there to this topic so that the OP can find my response.
  31. S

    MHB Can You Solve This Advanced Definite Integral Using Residue Theorem?

    This is classic one. Prove that $$ \int_0^\infty \frac{dx}{\left\{x^4+(1+2\sqrt{2})x^2+1 \right\}\left\{x^{100}-x^{99}+x^{98}-\cdots +1\right\}}=\frac{\pi}{2(1+\sqrt{2})}$$
  32. B

    Evaluate a Definite Integral to find Work

    Homework Statement Evaluate: from x=0 to x=1 and y=0 to y=1 ∫(y^2 + 2xy(dy/dx))dx and carry the integration out over x Homework Equations The Attempt at a Solution I know how to calculate double integrals with multiple variables but the (dy/dx) throws me off and it says to...
  33. L

    Evaluate definite integral. x if x<1; 1/x if x> or equal to 1.

    Evaluate definite integral. "x if x<1; 1/x if x> or equal to 1." 1. Consider the function: f(x) = {x if x<1 {1/x if x≥1 Evaluate the definite integral. ∫from 0 to 4 of f(x)dx 2. Okay, I think I vaguely remember something about these...
  34. D

    Can't solve this difficult definite integral for the life of me

    Homework Statement Definite integral: \int \frac{\ln(x+1)}{x^2+1} dx from x=0 to x=1 (sorry I don't know how to do integral boundaries with tex) The Attempt at a Solution I just am clueless on how to do this, I'm almost 100% sure you are not supposed to find the anti-derivative...
  35. S

    Proof of already solved Hard Improper Definite Integral

    Homework Statement Some friend of mine found this on a book: \int_{0}^{+inf}\frac{1-cos(\omega t)}{e^{\omega /C}(e^{\omega /T}-1)\omega }=ln[\frac{(\frac{T}{C})!}{|(\frac{T}{C}-iTt)!|}] The proof is left for the reader. Homework Equations The Attempt at a Solution First very safe step...
  36. S

    MHB Proving Definite Integral: \(\ln x/\sqrt{x(1-x^2)}\)

    Prove that \[\int_0^1 \frac{\ln x}{\sqrt{x(1-x^2)}}dx=-\frac{\sqrt{2\pi}}{8} \left(\Gamma\left(\frac{1}{4} \right)\right)^2 \] \(\Gamma (x)\) is the Gamma Function.
  37. J

    Minimizing a functional definite integral

    I have a definite integral defined by \begin{equation}T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g\end{equation} where G is a continuous function of a variable g, and g_{1} and g_{2} are known numbers. I want to minimize T\left(G\left(g\right)\right), that is I want to...
  38. P

    Definite integral of a quarter circle

    Homework Statement Find the definite integral of a quarter circle. Homework Equations x^2 +y^2=10 The Attempt at a Solution x^2=10-y^2 x=sqrt(10-y^2) ∫ sqrt(10-y^2)dy from 0 to sqrt(10) I'm not sure what to do here.
  39. P

    Definite integral with complex result?

    Homework Statement ∫ 3+ x√x from -1 to 4 Homework Equations The Attempt at a Solution ∫ 3+ x√x from -1 to 4 = 3x+(2(x^5/2))/5 evaluated from -1 to 4 (12 + (2(4^5/2))/5) +(3 +(2(-1)^5/2)/5) ? 15+64/5 +(2(-1^(5/2))/5) 139/5 -(2(-1^(5/2)))/5 139/5 -2i/5 is that right?
  40. P

    MHB Evaluation of definite integral

    I'm struggling for a long time to solve this integral $$\int_0^\infty e^{-x^2}cos(kx)dx$$ with $k>0$ I know there are a number of ways, but I'm interested in using complex integration. In particular, I believe that we can solve by integrating $e^{-z^2}$ over the boundary of the rectangule...
  41. M

    Expressing the limit of a sum as a definite integral

    Homework Statement Express the following as a definite integral: Express the attached limit as an integral. The Attempt at a Solution I have gotten as far as every part of the answer except the upper bound. the answer is: 10 ∫(from 1 to 10) [x-4lnx]dx 1 since the definition of...
  42. K

    Definite Integral with tricky deniminator

    Homework Statement I attached problem as a picture. Homework Equations I know that the integral of 1/x equals lnx if the derivative of the denominator is equal to the numerator. The Attempt at a Solution I tried to foil out the denominator and integrate by parts but it was very difficult...
  43. J

    MHB Definite Integral $\sqrt{\sin x}$: Answers

    $\displaystyle \int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx.\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}dx =$
  44. P

    Evaluate the definite integral

    Homework Statement Evaluate the definite integral from 0 to 18. ∫[x/(9+4x)^1/2]dx Homework Equations The Attempt at a Solution I know to use u-substitution and I set u = (9+4x)^1/2. I can't figure out where I'm messing up because I keep ending up with very large numbers which are...
  45. J

    MHB Definite Integral: $\int_{0}^{\frac{\pi}{2}}\frac{\cos x}{(a+b\cos x)^2}dx$

    $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x}{(a+b\cos x)^2}dx$
  46. A

    What are the units of a definite integral and its derivative?

    Hi, Suppose we have f(x) = x3. Integrating this function using the definite integral with the upper boundary being 3 and the lower boundary being 1 would result in 20. Does 20 have any units or is it unitless? Seeing how it is the area underneath a curve, I would imagine that it has square...
  47. P

    Find b, a and f(x) for the definite integral

    Homework Statement Three terms are used in a left hand sum to approximate the integral of ∫a to b f(x)dx ((2+0*(4/3))^2 * 4/3) + ((1+1*(4/3))^2 * 4/3) + ((2+2*(4/3))^2 * 4/3) find a possible value of b and a, and f(x Homework Equations Ʃ Δx(f(a+Δxi)) The Attempt at a Solution based...
  48. C

    Definite integral to indefine one

    Hi I would like to know what is the procedure to convert indefinite integral to definite one? For example I know ∫exp(-u^2)du from 0 to x is equal to ∫x*exp(-x^2*t^2)dt from 0 to 1 But I would like to know with what type of change of variable I get these?
  49. J

    Solving Definite Integral: x∫sqrt(5 + t^3) - e^t^2 dt

    Doing corrections on a test and I'm trying to solve this integral and I'm having quite a bit of trouble with it: __x Dx∫sqrt(5 + t^3) - e^t^2 dt __0 I tried solving it by breaking it up into two integrals: x________________x ∫sqrt(5 + t^3) dt - ∫e^t^2 dt 0________________0 Then I tried using...
  50. D

    Evaluating definite integral by substitution

    Homework Statement By using substitution u=\frac{1}{t}, or otherwise, show that \int^∞_1 \frac{t^5}{(1+t^3)^3}dt=\int^1_0 \frac{u^2}{(1+u^3)^3}du Homework Equations The Attempt at a Solution Well, the reverse can also be done (making t to u). However, I don't know how to...
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