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What’s the Geometrical interpretation of integrals? And what’s the physical significance of C?

What’s the Geometrical interpretation of integrals? And what’s the physical significance of C?

Grade:11

2 Answers

Vikas TU
14149 Points
4 years ago
Geometrical interpretation of indefinite integral

Let f (x) = 2x. Then ∫ f (x) dx = x2 + C. For different values of C, we get different integrals. But these integrals are very similar geometrically. 
Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis. 

Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin. The curve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along y-axis in positive direction. For C = – 1, y = x2– 1 is obtained by shifting the parabola y = x2 one unit along y-axis in the negative direction. Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis.

SREEKANTH
85 Points
4 years ago
 one unit along y-axis in the negative direction. Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis.2– 1 is obtained by shifting the parabola y = x2 one unit along y-axis in positive direction. For C = – 1, y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2, a parabola with its vertex on the origin. The curve y = x2. In this case, each integral represents a parabola with its axis along y-axis. Clearly, for C = 0, we obtain y = xindefinite integral + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the 2 + C. For different values of C, we get different integrals. But these integrals are very similar geometrically. Thus, y = x2Let f (x) = 2x. Then ∫ f (x) dx = x

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