Name: piecewise area between curves
Uploaded: 2026-06-30T08:00:37.538Z
Duration: 1 min 28 s
Channel: JoVE

Considérons une région délimitée par une fonction racine carrée, une fonction de racine cubique réfléchie et une fonction linéaire.La fonction linéaire intersecte les deux autres courbes, formant les frontières de la région.Pour trouver la surface, la région est divisée en deux segments basés sur le changement des courbes de frontière aux points d’intersection.Dans l’intervalle de x = 0 à x = 1, l’aire est trouvée en intégrant la distance verticale entre les courbes de la limite supérieure et inférieure par rapport à x, et en évaluant l’intégrale définie sur cet intervalle.Dans l’intervalle de x = 1 à x = 4, une paire différente de courbes frontières définit le haut et le bas de la région. L’aire sur ce segment est également déterminée en intégrant la distance verticale correspondante et en évaluant l’intégrale définie.L’aire totale enfermée est la somme des deux intégrales définies.Cette méthode est utilisée en économie du bien-être pour mesurer l’excédent économique total. Les courbes d’offre et de demande forment les frontières de la région. Le surplus total est calculé en intégrant la différence verticale entre les courbes de zéro à la quantité d’équilibre, représentant le bénéfice économique net.

aire entre courbes : résolution de problème

Area Between Curves: Problem Solving

Une r&#xE9;gion peut &#xEA;tre d&#xE9;limit&#xE9;e par trois courbes : une fonction racine carr&#xE9;e, une fonction racine cubique r&#xE9;fl&#xE9;chie et une fonction lin&#xE9;aire. La fonction lin&#xE9;aire intersecte chacune des deux autres courbes, et ces points d&#x2019;intersection d&#xE9;terminent o&#xF9; la limite de la r&#xE9;gion se modifie. &#xC9;tant donn&#xE9; que diff&#xE9;rentes courbes jouent le r&#xF4;le de fronti&#xE8;re sup&#xE9;rieure et de fronti&#xE8;re inf&#xE9;rieure selon les portions du graphique, l&#x2019;aire ne peut pas &#xEA;tre d&#xE9;termin&#xE9;e &#xE0; l&#x2019;aide d&#x2019;une seule mise en place sur l&#x2019;ensemble de l&#x2019;intervalle.
Pour calculer l&#x2019;aire, la r&#xE9;gion est d&#x2019;abord divis&#xE9;e en deux sous-intervalles au point d&#x2019;intersection o&#xF9; les courbes &#x22; sup&#xE9;rieure &#x22; et &#x22; inf&#xE9;rieure &#x22; changent de r&#xF4;le. Sur l&#x2019;intervalle de x = 0 &#xE0; x = 1, la courbe racine carr&#xE9;e est au-dessus de la courbe racine cubique r&#xE9;fl&#xE9;chie ; l&#x2019;aire sur cet intervalle est obtenue en int&#xE9;grant, par rapport &#xE0; x, la distance verticale entre les deux courbes, puis en &#xE9;valuant cette int&#xE9;grale d&#xE9;finie sur l&#x2019;intervalle. Sur l&#x2019;intervalle de x = 1 &#xE0; x = 4, la fonction lin&#xE9;aire devient l&#x2019;une des courbes de bord, et une autre paire de courbes d&#xE9;finit les bords sup&#xE9;rieur et inf&#xE9;rieur de la r&#xE9;gion. La m&#xEA;me proc&#xE9;dure est appliqu&#xE9;e : on int&#xE8;gre la distance verticale entre les courbes fronti&#xE8;res appropri&#xE9;es et on &#xE9;value l&#x2019;int&#xE9;grale d&#xE9;finie sur ce second intervalle.
L&#x2019;aire totale est ensuite obtenue en additionnant les deux r&#xE9;sultats. Le d&#xE9;coupage de la r&#xE9;gion au point de changement de fronti&#xE8;re garantit que chaque int&#xE9;grale repr&#xE9;sente une relation unique et coh&#xE9;rente de type &#x22; courbe sup&#xE9;rieure moins courbe inf&#xE9;rieure &#x22;, n&#xE9;cessaire pour une interpr&#xE9;tation g&#xE9;om&#xE9;trique pr&#xE9;cise.
Cette m&#xE9;thode de calcul d&#x2019;aire par morceaux est largement utilis&#xE9;e en &#xE9;conomie du bien-&#xEA;tre pour quantifier le surplus &#xE9;conomique total. Dans un diagramme de march&#xE9; classique, la courbe de demande et la courbe d&#x2019;offre forment les fronti&#xE8;res de la r&#xE9;gion du surplus, et le point d&#x2019;&#xE9;quilibre d&#xE9;termine o&#xF9; l&#x2019;intervalle pertinent se termine. Le surplus total est calcul&#xE9; en int&#xE9;grant la diff&#xE9;rence verticale entre la demande et l&#x2019;offre, de z&#xE9;ro &#xE0; la quantit&#xE9; d&#x2019;&#xE9;quilibre. L&#x2019;aire obtenue repr&#xE9;sente le b&#xE9;n&#xE9;fice &#xE9;conomique net g&#xE9;n&#xE9;r&#xE9; par les &#xE9;changes de march&#xE9;, en combinant les gains des consommateurs et des producteurs en une mesure unique.

Aire entre courbes : résolution de problème

Une r&#xE9;gion peut &#xEA;tre d&#xE9;limit&#xE9;e par trois courbes : une fonction racine carr&#xE9;e, une fonction racine cubique r&#xE9;fl&#xE9;chie et ...

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Applications of Integration

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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 the point where the upper and lower boundaries switch. From x = 0 to x = 1, the square root curve lies above the reflected cube root curve. The area for that part is found by integrating the vertical distance between those two curves with respect to x.

From x = 1 to x = 4, the linear function becomes part of the boundary. A different pair of curves then defines the upper and lower edges of the region. The same method is used again. The definite integral of the vertical distance is evaluated on the second interval, and the two results are added to get the total enclosed area.

Splitting the region at the boundary-change point keeps each integral in a simple upper-minus-lower form. That makes the geometric meaning of the area clear and accurate. This piecewise method also appears in welfare economics, where it helps measure total economic surplus. In a standard market diagram, the demand curve and supply curve form the bounds of the surplus region, and the equilibrium point marks the end of the interval. The total surplus is found by integrating the vertical difference between demand and supply from zero to the equilibrium quantity. The area represents the net economic benefit of market exchange and combines consumer and producer gains into one measure.

Piecewise Area Between Curves

area between curves: integrating with respect to x

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

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...

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 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 to split an area between curves into intervals, use upper minus lower integrals, and connect the method to total economic surplus in markets.

Piecewise 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

Piecewise Area Between Curves

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