# Integration and Differentiation

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 ?

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Why don't you try one of those free MOOC calc courses from Coursera? It's awesome. There's more to follow up. –  bonCodigo May 27 '14 at 22:51

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.

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Fiji? Seriously? Too cool : ) And plus one. –  Rudy the Reindeer Nov 3 '12 at 8:07
:) not so cool as it is hot ;) –  Ittay Weiss Nov 3 '12 at 8:29

It's most clear if you think of the derivative as the instantaneous rate of change, and an integral as a sum of all the tiny changes. $f'(t) dt$ is a tiny change in $f$ during a very short period of time. To get the total change in $f$ over an extended period of time, you add up all the tiny changes:

$$f(b) - f(a) = \int_a^b f'(t) \, dt.$$

In my opinion, the geometric interpretations of the derivative and integral (as slope of tangent line, and area under curve) are only secondary. The most basic thing to know about the derivative is that it's the instantaneous rate of change; the most basic thing to know about integration is that it's a way of adding up tiny changes to get the total change.

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Nice : "Some of all the tiny changes" - I would say all the instantaneous changes ;) –  bonCodigo May 27 '14 at 22:49