Name: estimating area with simpson’s rule
Uploaded: 2026-01-12T12:24:19.000Z
Duration: 1 min 26 s
Channel: JoVE

Os topógrafos frequentemente calculam a área de faixas de terra irregulares, como a região entre uma estrada reta e uma margem curva do rio.Como a fronteira curva do rio geralmente não possui uma equação matemática simples, é usada a Regra de Simpson.A regra de Simpson, baseada em um método de integração numérica, estima a área usando apenas algumas medidas amostradas.Aqui, essas amostras são distâncias perpendiculares da estrada até a margem do rio, medidas em três posições espaciadas igualmente.A Regra de Simpson funciona ajustando matematicamente uma parábola única através desses três pontos.A área sob a parábola é então encontrada integrando a equação quadrática ao longo desse intervalo.Após resolver a integral e aplicar os limites, a área total é obtida em termos de h e constantes.O ponto central nos permite encontrar imediatamente o termo constante, C. Substituir C nas expressões dos pontos externos e resolver essas expressões reduz a uma expressão em termos de y0, y 1 e y 2.Por fim, inserindo esses pontos em nosso resultado de integração, obtém a fórmula para a área estimada da faixa irregular de terra.

regra de simpson i

A Regra de Simpson &#xE9; um m&#xE9;todo de integra&#xE7;&#xE3;o num&#xE9;rica utilizado para aproximar o valor de uma integral definida quando uma antiderivada exata &#xE9; dif&#xED;cil ou imposs&#xED;vel de obter. O m&#xE9;todo estima a &#xE1;rea ajustando uma par&#xE1;bola &#xFA;nica aos tr&#xEA;s pontos equidistantes de uma curva e, em seguida, integrando a fun&#xE7;&#xE3;o quadr&#xE1;tica resultante ao longo do intervalo. Usando apenas um n&#xFA;mero reduzido de valores amostrados, a Regra de Simpson fornece uma aproxima&#xE7;&#xE3;o precisa de muitas fun&#xE7;&#xF5;es que variam suavemente.
Uma aplica&#xE7;&#xE3;o pr&#xE1;tica comum da Regra de Simpson surge na agrimensura, em que a &#xE1;rea de terrenos irregulares frequentemente precisa ser calculada. Os agrimensores podem precisar determinar a &#xE1;rea de uma faixa de terra entre uma estrada reta e a margem curva de um rio. Como a margem do rio geralmente n&#xE3;o segue uma equa&#xE7;&#xE3;o matem&#xE1;tica simples, a integra&#xE7;&#xE3;o anal&#xED;tica direta n&#xE3;o &#xE9; vi&#xE1;vel. Em vez disso, as dist&#xE2;ncias perpendiculares da estrada &#xE0; margem do rio s&#xE3;o medidas em tr&#xEA;s posi&#xE7;&#xF5;es equidistantes ao longo da faixa.
Essas dist&#xE2;ncias medidas servem como os pontos de dados usados para construir a par&#xE1;bola de aproxima&#xE7;&#xE3;o. A medida no ponto m&#xE9;dio permite que o termo constante da fun&#xE7;&#xE3;o quadr&#xE1;tica seja determinado diretamente. Substituindo esse valor nas express&#xF5;es para os pontos restantes, a equa&#xE7;&#xE3;o se reduz a uma express&#xE3;o que envolve apenas as dist&#xE2;ncias medidas, comumente denotadas por y_0, y_1 e y_2. Integrando a fun&#xE7;&#xE3;o quadr&#xE1;tica e aplicando os limites, obt&#xE9;m-se uma f&#xF3;rmula para a &#xE1;rea estimada em termos do espa&#xE7;amento entre as medidas e os valores amostrados, fornecendo um m&#xE9;todo pr&#xE1;tico para estimar &#xE1;reas de terrenos irregulares.

Regra de Simpson I

A Regra de Simpson &#xE9; um m&#xE9;todo de integra&#xE7;&#xE3;o num&#xE9;rica utilizado para aproximar o valor de uma integral definida quando uma ...

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Techniques of Integration

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Simpson's Rule I

estimating area with simpson’s rule

Simpson’s Rule is a numerical integration method for estimating the value of a definite integral. It is useful when an exact antiderivative is hard or impossible to find. The method works by fitting a unique parabola through three equally spaced points on a curve and then integrating that quadratic over the interval.

This approach can give a very accurate area estimate from only a few sampled values. It works well for many smoothly varying functions. That makes it a practical tool when a curve is too complex for direct integration.

A common use of Simpson’s Rule is surveying. Surveyors may need to find the area of a strip of land between a straight road and a curved riverbank. Because the riverbank does not usually follow a simple equation, analytical integration is not feasible.

To solve this problem, perpendicular distances from the road to the riverbank are measured at three equally spaced positions. These measurements become the data points used to build the parabola. The middle value helps determine the constant term of the quadratic function directly.

After substituting the measured values into the quadratic expressions, the equation depends on the three sample values, y0, y1, and y2. Integrating the quadratic and applying the limits gives a formula for the estimated area. The result is a practical way to estimate uneven land from a small set of measurements.

Estimating Area with Simpson’s Rule

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

Integration by parts is a calculus method for handling integrals that contain a product of two functions. It is useful when direct integration is not practical. The method comes from the product rule for differentiation, which says that the derivative of a product is the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating that product-rule identity and rearranging the terms, the integration by parts formula is obtained.

integration by parts: definite integrals

Using Parts and Substitution for Area

Integration by parts can make a definite integral easier to solve when the integrand is a product over a fixed interval. The method rewrites the problem as a product at the endpoints minus another definite integral that may be simpler to evaluate. A useful example is the definite integral of inverse tangent, or arctan x. Since there is no direct integration formula for arctan x, the integrand is written as arctan x times the constant function 1. The inverse tangent is chosen to differentiate,...

integration by parts: problem solving

Integration by Parts for Signal Processing

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integrals of powers of sine and cosine

Solving Trig Integrals with Odd and Even Powers

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integrals of powers of secant and tangent

integrals-of-powers-of-secant-and-tangent

Solving Secant and Tangent Integrals

Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function. When the power of secant is even, tangent is the better substitution variable. Since the derivative of tangent is secant squared, a factor of sec^2 x can be split off. The remaining even power...

trigonometric substitution

Finding Ellipse Area with Trig Substitution

Trigonometric substitution is a method for solving integrals that contain square root expressions from quadratic forms. It works well when the integrand looks like a familiar geometric equation, such as a circle or an ellipse. Here, the method is applied to estimate the area inside a Molniya satellite orbit. Molniya satellites move in highly elliptical orbits and repeatedly sweep across the same regions of space as they go around Earth. The orbit is modeled as an ellipse in standard form, and...

integration of rational functions using partial fractions

integration-of-rational-functions-using-partial-fractions

Splitting Rational Integrals Into Simpler Parts

Rational functions are ratios of two polynomials, and their integrals can be simplified by breaking them into smaller parts. A rational function is proper when the degree of the numerator is less than the degree of the denominator. When that is true, partial fraction decomposition rewrites the expression as a sum of simpler rational terms. To decompose a proper rational function, unknown constants are placed into the simpler terms. The denominator is then multiplied through to remove the...

rationalizing substitutions

Integrating Radicals with Rational Substitution

Integrating radicals can be easier when a rationalizing substitution turns the expression into a rational form. This method is useful when standard techniques do not work well because the integrand contains a radical, such as a cube root. The transcript uses a rod as a real-world example. Its linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Finding the total mass means integrating that density over the length of...

approximate integration

Riemann Sum Area Estimates

Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

trapezoidal rule

Estimating Distance from Velocity Data

Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements. The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid.

simpson's rule i

Simpson’s Rule is a numerical integration method for estimating the value of a definite integral. It is useful when an exact antiderivative is hard or impossible to find. The method works by fitting a unique parabola through three equally spaced points on a curve and then integrating that quadratic over the interval. This approach can give a very accurate area estimate from only a few sampled values. It works well for many smoothly varying functions. That makes it a practical tool when a curve...

simpson's rule ii

Simpson's Rule for Curved Sheet Area

Simpson's Rule helps estimate the area under a smooth curve, such as the curved profile of metal roofing sheets. In warehouse roofing, corrugated or curved sheets are used because they add strength, improve water drainage, and support better ventilation. To plan materials and design more accurately, engineers often need the area of these curved surfaces. The sheet shape can often be treated like a parabolic curve because it changes smoothly along its length. That makes numerical integration a...

improper integrals: infinite intervals

Improper Integrals and Exponential Decay

Improper integrals with infinite intervals appear when one limit of integration goes to positive or negative infinity. In that case, the area under the curve is unbounded, so ordinary definite integral methods do not apply directly. The integral is then defined by a limit, which helps show whether the total area is finite even when the interval is infinite. This idea is useful in exponential decay models. One example is the total integrated intensity of light moving through a uniform medium,...

improper integrals: discontinuous integrands

improper-integrals-discontinuous-integrands

Improper Integrals at an Endpoint

Improper integrals can appear when a definite integral has a discontinuity at one endpoint. The integrand may be undefined or become infinite there. In that case, the area under the curve is unbounded near the boundary. This behavior is often linked to a vertical asymptote at the edge of the interval. To evaluate this kind of integral, the endpoint is replaced with a variable. The integral is then written over an interval where the function is defined. After that, a limit is taken as the...

key terms and concepts

Learn how Simpson’s Rule uses three equally spaced measurements and a parabola to estimate definite integrals and irregular land area in surveying.

Estimating Area with Simpson’s Rule | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Estimating Area with Simpson’s Rule

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