11. Integration(1).

Today we begin writing about the second part of calculus – INTEGRATION. In the earlier posts, we discussed how differentiation helps us to divide our data into small components. This procedure helps us to get our results with exactitude. However, collecting and bringing these small pieces together is also mandatory to get the final result! The process that helps us to do this is called integration!

We have already studied in post #8 –

i) Differential Calculus – tells us the steepness of the graph at a point/rate of change of the function at a point.

ii) Integral Calculus – gives us an area under a particular region.

In the next few posts, we shall try to understand the concept of integral calculus and learn how it is used in conjunction with differential calculus.

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Integration

Integration is the act of bringing together smaller components into a single system.

To integrate means to join/unite small things to make one big thing. Thus, integration is the process of joining and blending small parts of something into a unified whole.

Integration is the reverse of differentiation.

In Differentiation ⇒ we break a system into smaller parts.
In Integration       ⇒ we bring the small parts together and find change over that time.

Thus, if the slope [dy/dx] of a function is known, the function itself can be found.

Notation for Integration

‘∫ ‘ is the symbol for integration. It is called the integral sign. It is an elongated ‘S’ for ‘SUM’ (In old German and English ‘S’ was written using this elongated shape).
∴ ∫ y.dx,     y → function.

e.g.  ∫ 2x.dx = x² + C. In this expression –

2x  ⇒ Integrand/function being integrated.
dx ⇒  This term tells us that ‘x’ is the variable w.r.t which we are integrating the function.
C   ⇒  Arbitrary constant of Integration.

The other notation is F(x) = ∫ f(x).dx    [Capital F(x) means integral of f(x)].

∴ F(x) → INTEGRAL
∴ f(x)→ DERIVATIVE

Why do we need the constant of integration ‘C’? Consider the following functions –

x²,  x² + 10,  x² -5.

If we differentiate these functions we get ,

d/dx (x² ) = 2x.

d/dx(x² + 10) = 2x+0 = 2x (as derivative of a constant is zero)

d/dx(x² -5) = 2x -0 = 2x.

When we reverse this i.e. we take the integral of 2x, we can’t determine the exact constant (it could be any number 10, -5 etc). Thus, while integrating, it is mandatory to introduce ‘C’ for indefinite integrals.

The following table gives the function and its integral –

Function f(x)	Indefinite Integral f(x).dx
constant, k	k x + C
x n	[(xn+1)/n+1] + C , n≠ – 1
1/x	ln |x| +C
ex	ex+ C
e-x	-e-x +C
ekx	(1/k) ekx +C
ax	ax/ln(a) +C
∫cf(x) dx, c= constant	c ∫f(x) dx
∫(f + g) dx	∫f dx + ∫g dx
∫(f – g) dx	∫f dx – ∫g dx

The above rules will be useful while using integration in our further study.

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The Process of Integration

The process of integration involves finding an area under a specific region/curve.This can be done as follows-

i) Divide the area/ region under the curve into small parts or rectangles.
ii) Find the sum of the areas of the rectangles.
iii) Find the precise area by finding the limit of the sum of the area of the rectangles, as the width of the rectangles tends to zero.

Sounds difficult? Let us try to understand this process with some examples.

Consider the graph below-

Suppose we want to find the area under the straight line between points 0 and x, in the graph above. We would first divide the area into many small rectangles and then keep decreasing the width of these rectangles as shown below-

As the width of the rectangles decreases, the number of rectangles increases. As we decrease the width of the rectangles, the area left out also decreases. This means we come closer to the real value of the area of the triangle. When the width is infinitesimally closer to zero (the last figure above), we are very close to the precise area of this triangle. This approximation gives us the exact area of the region under the curve for all practical purposes.

In the next post, we shall dive deeper into the process of integration. Till then,

Be a perpetual student of life and keep learning…

Good day!