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Area Under the Curve Formula

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Calculating the Area Under the Curve: Formula and Methods

Comprehensive Definition, Description, Examples & Rules 

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Introduction

Calculating the area under the curve is when you calculate the area between a curve and the -axis. It is the calculation of the area above the -axis or entirely below the -axis, or it also might be the combination of above and below the -axis. The area under the curve is between two points, which can be found by doing a definite integral between these two points. You can also calculate the area by giving integration at limits. 

Calculating the area under the curve is very important in various fields. These are:

  • You can have a visual integration of a particular area, which helps aerospace engineers find out how much space is enclosed by a curve in a particular area.
  • To calculate the area available inside a tank, you need to use this formula.
  • It also has an essential role in modern mathematics as calculus in geometry becomes very easy to value using the area under the curve formula.

What is the Area Under the Curve?

Calculating the area under the curve is generally used to find out the exact area of particular shapes that are not properly defined and are in a periodic motion that has an amplitude. The significance of finding the area under a curve in mathematics is:

  • It has an effective role in modern mathematics to calculate a geometrical equation.
  • Calculating the area of shapes that do not have a definite name happens by using the formula of the area under the curve.

The significance of finding the area under the curve in real-world applications is:

  • As discussed earlier, the main region where you calculate area under curve is finding space inside something. You can use it to calculate the cupboard’s area and available space.

‘Area under the curve’ refers to the area between a particular curve and the -axis. It is the visual representation of an integral and signifies the sum of all probabilities of a normal function.

Formula to Calculate Area Under Curve

The general formula that you might use for calculating the area under a curve is:

  • ∫ab f(x)dx

When you calculate it for two points, the area under a curve is found by doing a definite integral between the two points. If you break down the formula under the curve, then 

y = f(x) between x = a and x = b,

Integrate y = f(x) between a and b’s limits. 

You have to calculate the area by using the integrations with given limits. This is how you use the definite integrals in this context. 

To calculate the area under the curve, you have to break a size into numerous different angles, and the formula to calculate the total area under the curve is:

  • A = limx→∞∑ni=1f(x)

Methods of Calculation

The different methods by which you can calculate the area under the curve are:

Numerical Integration

The primary method of calculating the area under the curve is numerical integration. It is a method in which you can calculate the approximate value of an integral by using various numerical techniques. The different types of numerical techniques that you can use during the numerical integration are:

Trapezoidal Rule

It is the rule that you can use to find out the value of a definite integral by using the formula:

  • b∫a f(x) dx

It is the method in which you calculate the area under the curve by dividing the curve into small trapezoid shapes, and you sum the area of all these trapezoids, which will give you the total area under the curve. 

Simpson’s Rule

It is a rule you can also use to evaluate a definite integral for calculating the area under the curve. You use the anti-derivative of an integral to calculate the area by dividing it into different shapes. It is an extension of the trapezoidal rule and is similar to Newton’s division of a polynomial difference. The formula is:

  • ∫baf(x)dx≈h3[f(x0)+f(xn)+4×(f(x1)+f(x3)+…)+2×(f(x2)+f(x4)+…)]

Here, h = b – a / n, and n is the number of subintervals in even form. 

Analytical Integration

It is another method that you can use to calculate the area under the curve by using exact geometrical representations for circular or straight boundary segments. You can perform an analytical integration on circular and straight elements for which the point lies on a circular arc. 

The technique that the analytical integration uses is definite integrals. The formula for definite integrals is:

  • ∫baf(x)dx=F(b)−F(a) ∫ a b f ( x ) d x = F ( b ) − F ( a )

Here F'(x) = f(x).

Real-World Examples

Practical examples where you need to calculate the area under the curve are:

  • Calculate the total space available in a tank.
  • You can easily apply in physical science by calculating distance and time.
  • To calculate the entire area of a roller coaster, you need to use the area under the curve formula.
  • Calculating the hilly area road is also essential to use the area under the curve formula. 

The formulas that you can use are:

  • Tank: b∫a f(x) dx
  • Roller coaster: ∫baf(x)dx≈h3[f(x0)+f(xn)+4×(f(x1)+f(x3)+…)+2×(f(x2)+f(x4)+…)]
  • Hilly Roads: ∫baf(x)dx=F(b)−F(a) ∫ a b f ( x ) d x = F ( b ) − F ( a )

Applications

The practical applications to find out the area under a curve in different fields are:

  • Physics: It is the main section where you can use the area under the curve and calculate distance and time, and you can supplement the formula with the formula of Newton. 
  • Economics: You can use the area under the curve formula to calculate the consumer surplus as the formula can calculate the exact area in which the sales are happening.
  • Engineering: The most prominent area where you can use the area under the curve formula to find the volume and centroid, which will help calculate and make proper bridges and houses.

Case studies to define:

  • Journal of ICT 2022 by AN Tuah and AB Ibrahim
  • Journal of Biometrics 2022 by Willie Online Library

Challenges and Tips

The challenges are:

  • The formulas for calculating the area under the curve can be confusing, so understanding the formula’s usage should be your primary priority.

  • It can be challenging to determine the exact formula you have to use to calculate the area under the curve for the shape you want to calculate.
  • You can make common mistakes by using incorrect formulas and incorrect variables.

Tips

The tips that you can follow are:

  • Closely determine the shape for which you calculate the area under the curve.
  • Find the exact formula appropriate to calculate the area under the curve and do proper research before using the formula.
  • Understand the concepts very well before using the formula. 

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Key Takeaways

  1. You may need help in identifying the exact formula that you have to use to calculate the area under the curve.

  2. You use this formula to calculate the area of a shape that is not definite or has no value. 

  3. You need to identify and divide the shapes into different regions, calculate the specific area of a region, and then add them to get the total area.

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Frequently Asked Questions

The significance of finding the area under the curve is to find the accumulated amount of whatever the function is modelled into. You can find the area of shapes that are not properly defined in a particular manner, and you can find out the area of the shapes that are in periodic motion or amplitude.

The prerequisites for using the area under the curve formula are:

  • Finding the definite integral between the two points
  • The area under the x-axis will be negative
  • The area over the x-axis will be positive

The common challenges to face are:

  • The exact formula that is appropriate to find out the area of a shape.
  • Find out the curve’s equation when the exact definite integral is between the two points.
  • Understanding the type of formula and the exact concept that it is defining.

There are no details of particular online websites where you can find the calculations of area under the curve. You can search on Google and find your required answer, but in this blog, we have included a worksheet where you can help yourself find the calculations of the area under the curve. 

Taking the Edulyte worksheets, you can efficiently understand the resources to help you find the area under the curve calculations.

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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