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bezgin
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Why does the antiderivative give us the area? I can't really find it in textbooks.
bezgin said:Why does the antiderivative give us the area? I can't really find it in textbooks.
The antiderivative, also known as the indefinite integral, is used to find the area under a curve because it is the inverse operation of differentiation, which is used to find the slope of a curve. By integrating a function, we are essentially finding its "opposite" function, which helps us determine the area under the curve.
The Fundamental Theorem of Calculus states that differentiation and integration are inverse operations of each other. This means that the antiderivative is the reverse process of finding the derivative, which allows us to use the antiderivative to find the area under a curve.
No, the antiderivative can only be used to find the area under a curve if the given function is continuous and has a finite range. Additionally, the antiderivative can only be used to find the area between two points, not the total area under a curve.
The shape of the curve directly affects the use of antiderivatives to find the area. If the curve is constantly increasing or decreasing, the antiderivative can be used to find the exact area under the curve. However, if the curve has sharp turns or discontinuities, the antiderivative may not accurately represent the area under the curve.
No, the antiderivative can only be used to find the area under a curve, not the area of any arbitrary shape. It is important to note that the antiderivative can only be used to find the area under a continuous curve with a finite range, as mentioned before. Other shapes, such as triangles or circles, may require different methods to find their area.