Area under curve Definition and 37 Threads

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. M

    I Negative area above x-axis from integrating x^2?

    Suppose the following integration, ##\int_3^{-1} x^2 \, dx = \frac{1}{3}(-1)^3 - \frac{1}{3}(3)^3 = -\frac{28}{3}## However, if we have a look at the graph, The area between ##x = 3## and ##x = -1## is above the x-axis so should be positive. Dose anybody please know why the I am getting...
  2. Saracen Rue

    I How to evaluate the enclosed area of this implicit curve?

    The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below: In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
  3. brotherbobby

    Finding Area of Shaded Segment in Circle Using Calculus

    Problem Statement : To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP. Attempt 1 (without calculus): I mark some relevant lengths inside the circle, shown left. Clearly OS = 9 cm and SP = 12 cm using the Pythagorean...
  4. AntSC

    Acceleration-Time Graph and Velocity Multiple Choice Question

    The variation with time t of the acceleration a of an object is shown What is the change in velocity of the object from ##t=0## to ##t=6##? A. ##6ms^{-1}## B. ##8ms^{-1}## C. ##10ms^{-1}## D. ##14ms^{-1}## So apparently the answer is B, which I am having trouble reconciling. Using methods...
  5. brotherbobby

    The integral of a function ##f(x)## from its graph

    Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...
  6. S

    B Integration from "Area Under Curve" Perspective: Explained

    I can calculate the value of the integration, it will be ##\frac{\sqrt{3}}{2}## But if I draw the function and consider the area bounded by the curve and x-axis from x = 0.5 to x = 1, it seems that the area will be infinite because x = 1 is vertical asymptote. Why can't I consider from "area...
  7. karush

    MHB 4.1.306 AP Calculus Exam Area under Curve

    $\textsf{What is the area of the region in the first quadrant bounded by the graph of}$ $$y=e^{x/2} \textit{ and the line } x=2$$ a. 2e-2 b. 2e c. $\dfrac{e}{2}-1$ d. $\dfrac{e-1}{2}$ e. e-1Integrate $\displaystyle \int e^{x/2}=2e^{x/2}$ take the limits...
  8. T

    MATLAB Numerical approximation of the area under curve

    I am very new too Matlab and how it all works but I am having trouble understanding at what axis the numerical integration is occurring from on the graph that I plotted. So I am currently doing an experiment in gamma ray spectroscopy and due to issue with the software we found it hard to...
  9. I

    Find the minimum value of area under y = 4x - x^3

    Homework Statement Find the minimum value of the area of the region under the curve ## y=4x - x^3 ## from ##x=a## to ##x=a+1##, for all ##a>0##. This problem is from Stewart's Calculus Homework Equations Finding the area under the curve... The Attempt at a Solution I can set up the equation...
  10. orangeraindrops

    What does the area under a volume vs time graph represent?

    Homework Statement I have a function showing the volume of water in a bay at different times in the day, and I want to know what the area under this curve would represent (if it represents anything meaningful). I know how to integrate, so that isn't a problem. Homework Equations I am...
  11. R

    I Rate of change of area under curve f(x) = f(x)

    What lead to the equality of the, rate of change of area under curve f(x) = f(x). Was it, they were just compared(OR believed to be equal) and mathematically found to be equal. Or when one was integrated or differentiated the other appeared. Also I knew, integration was being used since...
  12. Z

    Simple integration to find area

    Homework Statement I wish to find the area under the curve y = 1/2^x between x=0 and x=1 but get an answer that is half the expected answer.Homework Equations Integrate y = 1/2^x to get -1/(2^x ln2) + Const This integration result was confirmed on Wolfram Slot in the range x = 0 to x = 1...
  13. M

    I Area under curve using Excel question

    Hi there, I was wondering if someone could help clarify something for me. I am using excel to find the area under a curve. I am using the : (B1+B2)/2*(A2-A1) equation to do it. However, due to the nature of the graph, all the value I am getting are negative. The values on the X axis decrease...
  14. S

    Stress-strain (area under curve)

    Homework Statement Deducing what the area under the stress-strain curve shows. There are four option in the attached image. I can discount work done by considering the units. The remaining ones seem plausible, but only one is true. Homework Equations stress = force / area; strain =...
  15. W

    Finding Area Under Curve: Rectangles vs Trapezia

    Two numerical methods for finding the area under a curve are the trapezium rule, where the area is split into trapezia, and the rectangle rule where you split into rectangles. The rectangle rule has two forms, one where you take the height at the midpoint and one where you take the height of the...
  16. O

    How to create a function such that area under curve is 250

    Suppose you have a given area under a curve, say 250, and want to come up with a function that produces this value. How would you do this? Although I came up with two basic functions as follows:First: Let y (x) = 5 from 0<x<50 , thus length*width = yx = 5*50 = 250. Second: Area of a...
  17. O

    Momentum, Impulse, Area under curve

    Δ≤ Homework Statement Find final velocity. Knowns: m = 4kg Initial v = 0 m/s F = Asin(xt) F = (2000N)(sin(\frac{1000π}{sec}*t) (0sec ≤ x ≤ .001sec) We need to find the impulse from a Force vs. Time graph. There is a preface to this problem that says if we work out the Force function...
  18. K

    Integral, why antiderivative is area under curve

    http://www.askamathematician.com/2011/04/q-why-is-the-integralantiderivative-the-area-under-a-function/ this website says f^\prime (c) (B-A) = f(B)-f(A) or f(c) (B-A) = F(B)-F(A) (since F’ =f). but seems like this is wrong. because B-A= Δ x, f'(c)*Δ x= Δy and the area might...
  19. A

    Find area under curve and energy of function (matlab)

    Homework Statement For the 1st one you wouldn't really need MATLAB I guess to find the area under the curve, it is 0 and so is its energy. For the 2nd one I got A=1.73 and so E=2.99.Homework Equations area under curve = evaluate integral from t=t1 to t=t2. in this case t=-2 to t=5 since they...
  20. D

    Graphing results of performance test and finding area under curve

    Problem statement, I am calculating the time it takes for my motorcycle to reach 55 miles per hour, through its engine Horsepower, Torque, transmission gearing, and sprocket gearing.(chain driven) I am using the equations: F=[mass/g]*[dV/dt] dt=[mass/g]*[dV/F] t=∫[mass/g]*[dV/F]...
  21. L

    Area under curve problem again

    Homework Statement Find the equation of the curve which passes through the point (-1,0) and whose gradient at any point (x,y) is 3x2-6x+4. Find the area enclosed by the curve, the axis of x and the ordinates x=1 and x=2. . The attempt at a solution I integrated and got the equation...
  22. L

    Another area under curve problem

    Homework Statement Sketch roughly the curve y = x^2(3-x) between x=-1 and x=4. Calculate the area bounded by the curve and the x-axis . The attempt at a solution I tried to find the area from x=-1 to x=4 I got 1 1/4 answer in the back of my textbook is 6 3/4 When i find the...
  23. L

    How Do You Calculate Areas Enclosed by Curves and Lines?

    Homework Statement Find the areas enclosed by the following curves and straight lines: c) y= (1/x2) -1 , y= -1 , x=1/2, and x=2 b) y = x3-1, the axes and y = 26 2. The attempt at a solution Okay I sketched the curve and to me it looks like the curve occupies no area at y=-1...
  24. B

    Area Under Curve: Find Intersection Points & Area

    Homework Statement Sketch the regions enclosed by the given curves. y = 3 cos 6x, y = 3 sin 12x, x = 0, x = π/12 Find the area as well.Homework Equations The sketch of the curve is given too which I uploaded. The Attempt at a Solution Trouble finding intersection points 3cos(6x)=...
  25. V

    Exploring Relationship of Limit & Integral Methods for Finding Area Under Curve

    Homework Statement I am in the process of studying integration and finding the areas under curves. So far, I know of two methods of finding the area under a curve: the limit method and the direct integral method. Could someone explain the relationship between these two methods? Homework...
  26. B

    Prove that area under curve by rectangle is less than integration

    hi all, I am suppose to compare the area of curve y=x2 with rectangles beneath that curve to show that, 1/2 + 1/3 + ...+ 1/n < log(n+1) i believe this some sort of harmonic series. Is there a way around this problem? Regards
  27. J

    Work - Area under curve of F vs. d graph

    Homework Statement Homework Equations W=FD Area under curve = Work The Attempt at a Solution Note, although the problem doesn't say this, and the given graph and description are deceptive, Force is NOT constant. I need to somehow find the Force required to move the cart from...
  28. H

    Calculating Area Under Curve with Trapezium Rule

    Homework Statement A modified form of the trapezium rule for calculating the area under a curve makes use of strips of varying width: by using narrower strips where the gradient varies more rapidly, better accuracy can be achieved. Create a function to perform the integral \int1/x dx between 1...
  29. J

    Integration and area under curve

    I have seen a thread with a similair title but passed up on what i want to know. I just want somebody to explain to me why definite integration equals the area under between the function and the x axis Ive just been through indefinite integration, then using the summation formulas in my AP...
  30. B

    Understanding the Riemann Integral and Its Application to Area Under Curves

    Homework Statement The question verbatim from the text is as follows: For every two-dimensional set C contained in R2 for which the integral exists, let Q(C) = ∫∫c x2 + y2 dxdy. If C1 = {(x,y): -1 ≤ x ≤ 1, -1 ≤ y ≤ 1}, C2 = {(x,y): -1 ≤ x=y ≤ 1}, C3 = {(x,y): x2 + y2 ≤ 1}, Find Q(C1), Q(C2)...
  31. R

    How Does One Calculate Specific Percentage Coordinates of an Area Under a Curve?

    Area under Curve REVERSED! ok now I have got a plot b/w x & y, which produced a curve using trapz function I evaluated the area now the important thing is that I have to take say 10% of the evaluated area & want to find at which coordinates 10% of the caculated area approaches to; similarly...
  32. G

    Antiderivative as Area under Curve

    The proof I'm familiar with for relating the antiderivative to the area under a curve involves usage of the mean value theorem, which for that particular case, implies continuity for the curve. Thus, integration as a process for finding the area under a curve should be valid under the...
  33. U

    Area under Curve: Find x-intercept & y-intercept

    Homework Statement Find the area bound by y= x - 2\sqrt{x} and y=0 The Attempt at a Solution X intercept = 4 Y intercept = 0 Using verticle rectangles.. dA = ((Upper curve)-(Lower curve)) dx \int(dA = \int(0-(x-2\sqrt{x})) dx A= -(1/2X^2)-((2x^(3/2))/(2/3)) Evaluated from...
  34. T

    Definite intergration area under curve bounded with line

    Homework Statement A cruve has the equation y = x{3} - 8x^{2} + 20x . The curve has stationary points A and B. There is a line through B parallel to y-axis and meets the x-axis at the point N. The region R is bounded by the curve , the x-axis and the line from A to N. Find the exact area under...
  35. F

    Area of Largest Rectangle in 1st Quadrant of y=4-x^2

    Homework Statement area of largest rectangle with sides parallel to axes (1st quad) under y=4-x^2 The Attempt at a Solution x(4-x^2) Area ` = (4-3x^2) x=0 and sqrt(4/3) The book says 4/3 (1.15) is wrong and that 3.08 is correct, how is this possible?
  36. M

    Calculus: Area Under Curve with Infinite Rectangles

    In calculus class when we were learning how definite integral derived from. We added up infinite # of rectangles under the curve. As n number of interval increased length of the base of the rectangles approached 0. Can you multiply 0 by infinity and get back the area you had.
  37. K

    Area of Region R & S in Quadrant 1: Calculating Integrals

    The question: Let R and S be regions in the first quadrant. The region R is bounded by the x-axis and the graphs of y=2-x^3 and y=tanx. The region S is bounded by the y-axis and the graphs of y=2-x^3 and y=tanx. a) Find the area of R b) Find the area of S I really need help starting...
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