The geometrical meaning of indefinite integration is that it represents a family of curves.
Understanding the Family of Curves
When you find the indefinite integral of a function, say (f(x)), you get a new function (F(x)) plus an arbitrary constant, usually denoted by (C). This result, (F(x) + C), is called the antiderivative of (f(x)). Geometrically, this antiderivative doesn't represent a single unique curve, but rather an entire collection or family of curves.
- Each value of the arbitrary constant (C) corresponds to a different member of this family of curves.
- These different members of the family are obtained by shifting one of the curves parallel to itself either vertically upwards or downwards along the y-axis.
Think of it like this: if (F(x)) is one particular antiderivative, then (F(x) + 1), (F(x) - 5), (F(x) + \sqrt{2}), etc., are all also antiderivatives. Each of these corresponds to the graph of (F(x)) shifted vertically by the value of (C).
The Role of the Constant of Integration (C)
The arbitrary constant (C) arises because the derivative of any constant is zero. Therefore, if (F(x)) is an antiderivative of (f(x)), then (F(x) + C) is also an antiderivative for any constant (C), because the derivative of (F(x) + C) is (F'(x) + 0 = F'(x) = f(x)).
Geometrically, this constant (C) acts as a vertical shift parameter for the graph of the antiderivative function.
Example: The Integral of 2x
Consider the indefinite integral of the function (f(x) = 2x).
The indefinite integral is:
[ \int 2x \, dx = x^2 + C ]
Here, (F(x) = x^2) and (C) is the arbitrary constant. This integral represents a family of parabolas.
As highlighted in the reference, for the integral of 2x, each integral represents the parabola with its axis along the y-axis. Different values of (C) give different parabolas within this family:
- If (C = 0), the curve is (y = x^2).
- If (C = 1), the curve is (y = x^2 + 1) (the basic parabola shifted up by 1 unit).
- If (C = -2), the curve is (y = x^2 - 2) (the basic parabola shifted down by 2 units).
| Value of C | Equation | Geometric Representation |
|---|---|---|
| ... | ... | ... |
| -2 | (y = x^2 - 2) | Parabola shifted down by 2 units |
| -1 | (y = x^2 - 1) | Parabola shifted down by 1 unit |
| 0 | (y = x^2) | Basic parabola through the origin |
| 1 | (y = x^2 + 1) | Parabola shifted up by 1 unit |
| 2 | (y = x^2 + 2) | Parabola shifted up by 2 units |
| ... | ... | ... |
These are all parabolas of the form (y = x^2 + C), and they are all vertical translations of each other. This illustrates the core geometrical meaning: the indefinite integral generates a family of curves related by vertical shifts.
In summary, as stated in the reference, the geometrical interpretation of indefinite integral is that different values of arbitrary constants will correspond to different members of the family, which can be obtained by shifting one of the curves parallel to itself.
