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

Rozważmy obszar ograniczony funkcją pierwiastka kwadratowego, funkcją pierwiastka sześciennego odbitym oraz funkcją liniową.Funkcja liniowa przecina pozostałe dwie krzywe, tworząc granice tego obszaru.Aby wyznaczyć powierzchnię, obszar dzieli się na dwa segmenty w zależności od zmiany krzywych brzegowych w punktach przecięcia.W przedziale od x = 0 do x = 1, pole znajduje się przez całkowanie pionowej odległości między górną a dolną krzywą brzegową względem x oraz obliczanie całki określonej na tym przedziale.W przedziale od x = 1 do x = 4 inna para krzywych brzegowych definiuje górę i dół obszaru. Pole nad tym odcinkiem jest również wyznaczane przez całkowanie odpowiadającej odległości pionowej i wyliczenie całki określonej.Całkowita powierzchnia zamknięta jest sumą dwóch całek oznaczonych.Metoda ta jest stosowana w ekonomii ogólnej nadwyżki gospodarczej do pomiaru całkowitej nadwyżki ekonomicznej. Krzywe popytu i podaży wyznaczają granice regionu. Całkowita nadwyżka oblicza się przez całkowanie pionowej różnicy między krzywymi od zera do wielkości równowagowej, co reprezentuje netto korzyść ekonomiczną.

obszar między krzywymi: rozwiązywanie zadań

Area Between Curves: Problem Solving

Obszar mo&#x17C;e by&#x107; ograniczony przez trzy krzywe: funkcj&#x119; pierwiastka kwadratowego, funkcj&#x119; odbitego pierwiastka sze&#x15B;cianowego oraz funkcj&#x119; liniow&#x105;. Funkcja liniowa przecina ka&#x17C;d&#x105; z pozosta&#x142;ych dw&#xF3;ch krzywych, a punkty przeci&#x119;cia okre&#x15B;laj&#x105;, gdzie zmienia si&#x119; granica obszaru zamkni&#x119;tego. Poniewa&#x17C; r&#xF3;&#x17C;ne krzywe pe&#x142;ni&#x105; rol&#x119; granic g&#xF3;rnej i dolnej w r&#xF3;&#x17C;nych cz&#x119;&#x15B;ciach wykresu, obszaru nie mo&#x17C;na obliczy&#x107; przy u&#x17C;yciu jednego zapisu ca&#x142;kowego na ca&#x142;ym przedziale.
Aby obliczy&#x107; pole, obszar najpierw dzieli si&#x119; na dwa segmenty w punkcie przeci&#x119;cia, w kt&#xF3;rym zmieniaj&#x105; si&#x119; krzywe graniczne &#x201E;g&#xF3;rna&#x201D; i &#x201E;dolna&#x201D;. W przedziale od x = 0 do x = 1 krzywa pierwiastka kwadratowego le&#x17C;y powy&#x17C;ej krzywej odbitego pierwiastka sze&#x15B;cianowego, wi&#x119;c pole w tym segmencie oblicza si&#x119; przez ca&#x142;kowanie odleg&#x142;o&#x15B;ci pionowej mi&#x119;dzy tymi dwiema krzywymi wzgl&#x119;dem x, a nast&#x119;pnie obliczenie warto&#x15B;ci tej ca&#x142;ki oznaczonej na ca&#x142;ym przedziale. W przedziale od x = 1 do x = 4 funkcja liniowa staje si&#x119; jedn&#x105; z granic obszaru, a inna para krzywych definiuje g&#xF3;rn&#x105; i doln&#x105; kraw&#x119;d&#x17A; obszaru. Stosuje si&#x119; t&#x119; sam&#x105; procedur&#x119;: ca&#x142;kowanie pionowej odleg&#x142;o&#x15B;ci mi&#x119;dzy odpowiednimi g&#xF3;rnymi i dolnymi krzywymi granicznymi oraz obliczanie warto&#x15B;ci ca&#x142;ki oznaczonej dla tego drugiego przedzia&#x142;u.
Nast&#x119;pnie oblicza si&#x119; ca&#x142;kowity obszar, dodaj&#x105;c oba wyniki. Podzia&#x142; obszaru w punkcie zmiany granicy zapewnia, &#x17C;e ka&#x17C;da ca&#x142;ka reprezentuje pojedyncz&#x105;, sp&#xF3;jn&#x105; zale&#x17C;no&#x15B;&#x107; &#x201E;g&#xF3;rna minus dolna&#x201D;, co jest niezb&#x119;dne do dok&#x142;adnej interpretacji geometrycznej.
Ta metoda obliczania obszaru jest szeroko stosowana w ekonomii dobrobytu do okre&#x15B;lania ca&#x142;kowitej nadwy&#x17C;ki ekonomicznej. Na standardowym diagramie rynkowym krzywa popytu oraz krzywa poda&#x17C;y wyznaczaj&#x105; granice obszaru nadwy&#x17C;ki, a punkt r&#xF3;wnowagi okre&#x15B;la, gdzie ko&#x144;czy si&#x119; dany przedzia&#x142;. Ca&#x142;kowit&#x105; nadwy&#x17C;k&#x119; oblicza si&#x119;, ca&#x142;kuj&#x105;c pionow&#x105; r&#xF3;&#x17C;nic&#x119; mi&#x119;dzy popytem a poda&#x17C;&#x105; od zera do wielko&#x15B;ci r&#xF3;wnowagi. Otrzymany obszar reprezentuje korzy&#x15B;&#x107; netto generowan&#x105; przez wymian&#x119; rynkow&#x105;, &#x142;&#x105;cz&#x105;c korzy&#x15B;ci konsument&#xF3;w i producent&#xF3;w w jednej mierze.

Obszar między krzywymi: Rozwiązywanie zadań

Obszar mo&#x17C;e by&#x107; ograniczony przez trzy krzywe: funkcj&#x119; pierwiastka kwadratowego, funkcj&#x119; odbitego pierwiastka sze&#x15B;cianowego ...

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Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Applications of Integration

Zastosowania całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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