while learning Calculus at College level mathematics classes, we were told that:
Differentiation and Integration are opposite or complementary to each other….(1)
Differentiation is Tangent to the given curve ….(2)
and Integration is Area under curve….(3)
Now my question is: From (1), (2) and (3) above, what is complementary or opposite in Tangent to curve and Area under the curve ?
Best Answer
Very loosely speaking integration and differentiation are complementary in the following sense (The Fundamental Theorem of Calculus). Given a continuous function $f:[a,b]\to \mathbb {R}$ define $F(t)=\int _a^tf(x)dx$. Then $F'(x)=f(x)$ holds for all $x\in [a,b]$. Stated differently, if $F:[a,b]\to \mathbb {R}$ is differentiable then $\int _a^bF'(x)dx=F(b)-F(a)$. So, in some sense, the integral of the derivative is the original function and, in some sense, the derivative of the integral is the original function.
This is typically what is meant in calculus courses when saying that the derivative and integral are complementary. There is a nice geometric interpretation of this theorem (and its proof) that considers the relation between the area under the curve and the slopes of the curve.
However, one needs to remember that this complimentarity should be taken with a grain of salt. The derivative of a function is, if it exists, a single function while the indefinite integral of a function, if it exists, is a whole family of functions (e.g., for $f(x)=2x$ the derivative is $f'(x)=2$ while $\int f(x)dx$ is the family of functions $x^2+C$). In generalizations to higher dimensions the correct interpretation of the derivative at a point is as a linear transformation while the Riemann integral is a direct generalization of the notino of area to the notion of volume. The relation between derivatives and integrals in higher dimensions is much more subtle and deep, culminating in Stokes' Theorem (preferably using differential forms). In some sense one can say the complimentarity does not quite hold in higher dimensions the way it holds in dimension 1.