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. R

    Solve Definite Integral Homework: \int_0^a {x\sqrt{x^2+a^2}\,dx}

    Homework Statement \int_0^a {x\sqrt{x^2+a^2}\,dx} Also, (A>0) The Attempt at a Solution Firstly, I set u=x^2+a^2 Then take the derivative, du=2x dx 1/2\int_0^a {\sqrt{u}\,du} Now I integrated. So (1/3) * [(x^2+a^2)^3/2] from a to 0. I ended up with...
  2. S

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  3. S

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  4. M

    Definite Integral of x^2*sqrt(7x+9) from 0 to 1 - Need Help!

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  5. K

    Please HeLp with definite integral

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  6. L

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  7. Saladsamurai

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  8. D

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  9. S

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  10. A

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  11. W

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  12. D

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  13. D

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  14. M

    Solving Definite Integral Problem with Density Function f(x) = 2e^(-0.25x)

    Hello, Okay so I was given a density function: f(x) = 2e^(-0.25x) The problem asks for the value of Pr(X < or = 3) I first figured out the probability density function first by let \int (3,0) k.25e^(-.25x) = 1 And figured out that k = .45 and continue solving my...
  15. S

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  16. S

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  17. R

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  18. S

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  19. C

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  20. A

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  21. G

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  22. V

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  23. S

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  24. A

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  25. S

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  26. B

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  27. B

    Integrating 110*1.10^(t) from 0 to 35: Solution

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  28. O

    How do I evaluate this definite integral in Calc 1?

    I am trying to evaluate this integral, \[ \int_{ - r}^r {\left( {s\sqrt {1 + \frac{{x^2 }}{{r^2 - x^2 }}} } \right)} {\rm }dx\] Is it a valid integral? If I evaluate it by plugging in r and -r, it becomes undefined. How else can I evaluate this integral? I'm in calculus 1 and I am...
  29. C

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    hi guys yeh, I am still going through revision, and I am also stuck on this question. *integral sign*(upper limit 4, lower limit 2) x/sqrt(3x^2 + 4).dx When i look at this, i think of letting u = 3x^2 + 4, then bringing that sqrt to the top,solving for dx, and then find the integral, and...
  30. S

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  31. E

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  32. maverick280857

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  33. C

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    If I want to evaluate: \int^1_0 \frac {dx}{(x+1)^2} I need to use the Fundamental Theorem of Calculus right? SO wouldn't I have to solve \int^b_a \frac{dx}{(x+1)^2} = \frac{(x+1)^3}{3} = \frac {8}{3} - \frac {1}{3} ? But the answer is \frac {1}{2}
  34. maverick280857

    Solving Definite Integral with Floor Function: Help Needed

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  35. L

    Can You Tackle This Tricky Definite Integral Challenge?

    Who can solve or give hint for the following integral? \int_0^4 3x \sqrt{5^2-x^2}dx :-p :eek:
  36. D

    Using a Power Series to Approximate a Definite Integral

    I think I got pretty close to the answer to this problem. However, I just can't obtain the right approximation at the end. Please, help me find where I made a mistake. Thanks. -------------------------------------------------------------- Use a power series to approximate the definite...
  37. Q

    Solving Definite Integral Problem: Average Velocity of Ball Dropped from Rest

    Please help me! I got stuck on this problem: (1) A ball is dropped from rest, and after t seconds its velocity is v ft/sec. Neglecting air resistance, show that the average velocity during the first T/2 seconds is 1/3 of the average velocity during the next T/2 seconds. Will I integrate v? If...
  38. C

    Is the Antiderivative of 1/x Actually ln(x)?

    ive been trying to do this problem and its annoying! The function to be integrated: 1 + 1/x + x dx Interval: [8,2] when anti differentiating the fuction i get x + x^2/2 + lnx but i don't...
  39. M

    Any proof for the definition of the definite integral

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  40. M

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    Hi, Physics books gloss over math. Sometimes it bothers me. Given a separable function of time and position f = h(x)g(t) then d / dt of [inte] h(x)g(t)dx = [inte] h(x) dg(t)/dt dx Where the derivative in the second integral is a partial deriv. Why the chain rule does not apply...
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