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Engg. Maths - Calculus - Concept Card 3D

ENGG. MATHS - CALCULUS - CONCEPT CARD 3D

3D - INTEGRATION AND ITS APPLICATION

Integrals:

The following are the two types of integrals:

1. Definite Integrals:

Signed Area:

A curve above the x-axis is said to enclose positive area, while a curve below the x-axis is said to enclose a negative area.

Definite Integrals:

The definite integral  is the signed area between the function  and the  from x=a to x=b.

2. Indefinite Integrals:

Anti Derivatives:

The function F(x) is an antiderivative of f(x) if .

Indefinite Integrals:

The indefinite integral  represents a family of antiderivatives: 

Fundamental Theorem of Calculus:

1. If f(x) is continuous on the interval [a,b], then the area function  is continuous and differentiable on the interval and .

2. If f(x) is continuous on the interval [a,b] and F(x) is any antiderivative of f(x), then .

Approximations to the Area:

1. Left Hand Rectangle Approximation:

a. The area under f(x) is approximated by a series of n rectangles:

i.e. the interval [a,b] is divided into n equal sub-intervals of width   to obtain (n+1) points on the x-axis at:

b. Now the 'n' points,

are the values of the function evaluated at the bottom left hand corners of the n rectancgles, which are also taken as the height of the corresponding rectangles.

c. The total area of n rectangles then is: 

2. Right Hand Rectangle Approximation:

a. The area under f(x) is approximated by a series of n rectangles:

i.e. the interval [a,b] is divided into n equal sub-intervals of width   to obtain (n+1) points on the x-axis at:

b. Now the 'n' points,

are the values of the function evaluated at the bottom right hand corners of the n rectancgles, which are also taken as the height of the corresponding rectangles.

c. The total area of n rectangles then is: 

Note:

3. Midpoint Rule:

The height of each rectangle can be taken to be the value of the function evaluated at the midpoint of each rectangle.

The total area of n rectangles is given by:

4. Trapezoidal Rule:

The area of a trapezoid is 1/2 X (sum of the parallel sides) X (distance between them).

The total area of n trapezoids is given by:

5. Simpson's Rule:

The total area under f(x) is given by:

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