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# Integral Calculus

## Integral Calculus PDF Notes, Important Questions and Formulas

Indefinite Integration

Introduction:

Both part of the fundamental Theorem establish connections between antiderivatives and definite integrals.

Part 1 says that if is continuous, then  dt is an

Antiderivatives of f. part 2 says that  dx can be

Found by evaluating F(b) – F(a), where F is an antiderivative of f. We need a convenient notation for antiderivatives that makes them easy to work with. Because of the relation given by the Fundamental Theorem between antiderivatives

and integrals, the natation  dx is traditionally used for

Antiderivative of f and is called as indefinite integral.

Thus   Means F ‘(x) = f(x)

For Example, we can write

because

So we can regard an indefinite integral as representing an entire family of function (one antiderivative for each value of the constant C). You should distinguish carefully between definite and

Indefinite integrals. A definite integrals  is a

Number, whereas an indefinite integral  is family

of functions. The connection between them is given by part 2 of the Fundamental Theorem. If f is continuous on [a, b] then

The effectiveness of the Fundamental Theorem depends on having a supply of antiderivatives of functions. We adopt the convention that when a formula for a general indefinite integral is given, it is valid only on an interval. Thus, we write

With the understanding that it is valid on the interval  or on the interval. This true despite the fact that the general antiderivatives of the function  is

Definite Integration

THE AREA PROBLEM

Use rectangles to estimate the area under the parabola y = x2 from, 0 to 1

We first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips S1, S2, S3, and S4 by drawing the vertical lines x = ¼

X=  and x=  as in figure (a)

We can approximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip [see Figure (b)]. In other words, the heights of these rectangle are the values of the function

F(x) = x2 at the right end points of the subinterval

Each rectangle has width  and the heights are  ,

. If let R4 be the sum of the areas

of these approximating rectangle,

we get,

From the Figure (b) we see that the area A of S is less than

R4 , So

A < 0.46875

Instead of using the rectangles in Figure (b) we could use the smaller rectangles in Figure (c) whose heights are the values of f at the left endpoints of the subintervals. (The leftmost rectangle has collapsed because its heights is 0). The sum of the areas of these approximating rectangles is

We see that the area of S is larger than L4, so we have lower and upper estimates for A

0.21875 < A < 0.46875

We can repeat this procedure with a larger number of strips. Figure (d), (e) shows what happens when we divide the region S into eight strips of equal width.

(d) Using left endpoint

(e) Using right endpoint

By computing the sum of the areas of the smaller rectangles (L8) and the sum of the areas of the larger rectangles (R8), we obtain better lower and upper estimates for A : 0.2734375 < A < 0.3984375 So one possible answer the question is to say that the true area of S lies somewhere between 0.2734375 and 0.3984375. We could obtain better estimates by increasing the number of strips.

PROPERTIES OF DEFINITE INTEGRAL

P-1: CHANGE OF VARIABLE

The definite integral  is a number, it does not

Depend on x. in fact, we could use any latter in place of x without changing the value of the integral;

P-2: CHANGE OF LIMIT

When we define the infinite integral  we

Implicity assumed that a < b. But the definition as a limit of sum makes sense even if a > b. Notice that if we reverse a and b, then x changes from (b – a)/n to (a – b)/n.

Therefore,

P-3: ADDITIVITY WITH RESPECT TO THE INTERVAL OF INTEGRATION:

This not easy to prove to in general, but for the case where

and a < c < b Property 7 can be seen from the

Geometric interpretation in Figure: The area under y= f(x)

from a to c is equal to the total area from a to b.

Area Under the Curve

CALCULATING AREA BY USING

HORIZONTAL STRIP

AREA BY HORIZONTAL ATRIPS

To determine the area bounded by the curve x=f(y), the y-

axis and abscissa at y=c & y=d is

i.e.

CALCULATING AREA BY USING

VERTICAL STRIP

AREA BY VERTICAL STRIPS

To determinant area bounded by curve y=f(x), the x-axis

and the ordinates at x= a & x= b is

Case-I

If y=f(x) lies completely above the x-axis

i.e.  A=

Case –II

If y=f(x) lies completely below the x-axis then A is negative.

The convention is to consider the magnitude only

i.e.