Name: vertical slices for area between curves
Uploaded: 2026-07-24T11:53:29.280Z
Duration: 1 min 21 s
Channel: JoVE

Considere duas funções contínuas em um intervalo fechado [a,b]. A região &#55349;&#56390; está situada entre as curvas, delimitada verticalmente pelas funções e horizontalmente por a e b.A área entre as curvas pode ser estimada dividindo a região em n faixas verticais de largura igual. Cada tira é tratada como um retângulo, que é usado no método de soma de Riemann.Considere ai-ésima tira. Pegue a base e depois a altura da tira em xi. A área desse retângulo é calculada multiplicando sua altura por sua base.A área total estimada dessa curva é calculada somando as áreas de todos os n retângulos. Esses retângulos não cobrem exatamente toda a área, mas à medida que o número de retângulos aumenta e n se aproxima do infinito, a área total se aproxima da área exata.Em vez de somar as áreas dos retângulos, a integração em relação a x resulta nas áreas de fatias verticais infinitamente finas, fornecendo a área exata entre as curvas.Esse conceito é usado em economia para encontrar a lacuna de desigualdade, que é a área entre a linha de igualdade perfeita e a curva da distribuição real da renda.

Área entre as curvas: integração em relação a x

Area Between Curves: Integrating With Respect to x

Considere duas fun&#xE7;&#xF5;es cont&#xED;nuas definidas em um intervalo fechado [a, b]. A regi&#xE3;o entre essas curvas &#xE9; limitada verticalmente por seus gr&#xE1;ficos e horizontalmente pelos pontos extremos do intervalo. O objetivo &#xE9; medir a &#xE1;rea dessa regi&#xE3;o.Uma estimativa inicial da &#xE1;rea pode ser obtida dividindo o intervalo em um grande n&#xFA;mero de tiras verticais estreitas de igual largura. Cada tira &#xE9; aproximada por um ret&#xE2;ngulo cuja altura &#xE9; dada pela diferen&#xE7;a vertical entre as duas fun&#xE7;&#xF5;es em um ponto escolhido dentro da tira. A soma das &#xE1;reas desses ret&#xE2;ngulos resulta em uma aproxima&#xE7;&#xE3;o da regi&#xE3;o por uma soma de Riemann. Esse processo &#xE9; representado por\begin{equation*}\text{Area} = \lim_{n \to \infty} \sum_{\textit{i}=1}^{\textit{n}} [ f(x_i) - g(x_i)] \Delta x\end{equation*}&#xA0;Como os ret&#xE2;ngulos n&#xE3;o coincidem exatamente com as fronteiras curvas, essa soma fornece apenas uma aproxima&#xE7;&#xE3;o quando um n&#xFA;mero finito de tiras &#xE9; utilizado. No entanto, &#xE0; medida que o n&#xFA;mero de tiras aumenta e suas larguras diminuem, a aproxima&#xE7;&#xE3;o melhora. No caso-limite, os ret&#xE2;ngulos tornam-se infinitamente finos, e a soma converge para a &#xE1;rea exata.Esse processo de passagem ao limite substitui a soma de Riemann por uma integral definida, que calcula a &#xE1;rea por meio de fatias verticais infinitamente finas. A &#xE1;rea exata entre as curvas &#xE9; dada por\begin{equation*}\text{Area} = \int_a^b [ f(x) - g(x) ]\,dx\end{equation*}&#xA0;Esse m&#xE9;todo para encontrar a &#xE1;rea entre curvas tem aplica&#xE7;&#xF5;es importantes al&#xE9;m da matem&#xE1;tica pura. Em economia, por exemplo, a &#xE1;rea entre a linha de igualdade perfeita e a curva que representa a distribui&#xE7;&#xE3;o real de renda &#xE9; usada para quantificar a desigualdade de renda. O tamanho dessa &#xE1;rea fornece uma medida num&#xE9;rica da lacuna de desigualdade, demonstrando como ideias geom&#xE9;tricas do c&#xE1;lculo podem ser aplicadas &#xE0; an&#xE1;lise econ&#xF4;mica do mundo real.

Considere duas fun&#xE7;&#xF5;es cont&#xED;nuas definidas em um intervalo fechado [a, b]. A regi&#xE3;o entre essas curvas &#xE9; limitada verticalmente ...

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Calculus

Cálculo

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Applications of Integration

Aplicações da Integração

KEY TERMS AND CONCEPTS

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vertical slices for area between curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup.

A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point inside the strip.

Adding the areas of all the rectangles gives a Riemann sum, which is a way to approximate area with many small pieces. Because the rectangle edges do not exactly follow the curves, the result is only an estimate when the number of strips is finite. As the strips become more numerous and thinner, the approximation improves.

In the limiting case, the strips become infinitely thin. Then the Riemann sum becomes a definite integral, which gives the exact area using infinitely thin vertical slices. This method is useful beyond pure math. In economics, the area between the line of perfect equality and a curve showing actual income distribution can be used to measure income inequality. The size of that area gives a numerical value for the inequality gap.

Vertical Slices for Area Between Curves

area between curves: integrating with respect to x

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup. A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point...

area between curves: integrating with respect to y

area-between-curves-integrating-with-respect-to-y

Horizontal Slices for Area Between Curves

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips. The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance...

area between curves: problem solving

Piecewise Area Between Curves

Piecewise area between curves uses more than one integral to find the size of a region on a graph. In this example, the region is enclosed by a square root function, a reflected cube root function, and a linear function. The line intersects each of the other two curves, and those intersection points show where the boundary of the region changes. The area cannot be found with one setup over the full interval because the top and bottom curves are not the same everywhere. The region is split at...

calculation of volume of solids by integration

calculation-of-volume-of-solids-by-integration

Slicing Methods for Solid Volume

Slicing methods help find the volume of solids with changing cross-sections. A basic solid, like a rectangular box, is easy to measure because its base area stays the same along its height. Many real objects do not have that regular shape, so elementary geometry is not enough. The slicing method solves this by breaking a solid into many very thin slices. These slices are taken perpendicular to a chosen axis, often the x-axis. Each slice has a small thickness, written as delta x, and a...

volumes of solids of revolution

Disk Method for Rotated Region Volume

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering. When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be...

finding volume using cross-sectional area

finding-volume-using-cross-sectional-area

Tetrahedron Volume from Triangle Slices

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available. A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the...

problem solving: volume

Finding the Volume of a Jet Fuel Tank

The volume of a jet fuel tank can be found with solids of revolution. The tank is modeled by rotating a two-dimensional region around the x-axis. This creates a three-dimensional shape that is symmetric about the axis of rotation. The region runs from 0 to 2 meters, so the tank’s shape can be measured along that length. Because the boundary curve touches the axis, the disk method works well here. The disk method breaks the solid into many very thin circular slices that are perpendicular to the...

average value of a function

Estimating Function Behavior with Average Value

The average value of a function gives one number that summarizes how a quantity changes over a closed interval. It can also be viewed as the height of a rectangle whose area matches the net area under the curve. That net area includes both positive and negative parts of the function, so the result reflects the function’s overall behavior. A greenhouse temperature over a 24-hour period is a practical example of this idea. The temperature may rise and fall throughout the day, but it is still...

key terms and concepts

Learn how vertical strips and Riemann sums lead to a definite integral for finding area between curves, plus a real-world example from economics.

Vertical Slices for Area Between Curves | 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

Vertical Slices for Area Between Curves

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