SHORT NOTES: Definite Integral

Definition:

If f(x) is continuous in the interval [a,b] and g(x) is an antiderivative of f(x)(i.e ${\color{Blue} \frac{d}{dx}g(x)=f(x)}$ ),then

$\int_{a}^{b}f(x)dx=g(b)-g(a)$

This is called as fundamental theorem of integral calculus and it establishes a relation between definite integral and indefinite integral.

So to evaluate definite integral first evaluate indefinite integral then put the limits and find the difference.

Properties:

(i) $\int_{a}^{b}f({\color{Blue} x})d{\color{Blue} x}=\int_{a}^{b}f({\color{Red} y})d{\color{Red} y}$ Independent of symbol of variable

(ii) $\int_{{\color{Blue} a}}^{{\color{Red} b}}f(x)dx=-\int_{{\color{Red} b}}^{{\color{Blue} a}}f(x)dx$
(iii) $\int_{a}^{b}f(x)dx=\int_{a}^{{\color{Red} c}}f(x)dx+\int_{{\color{Red} c}}^{b}f(x)dx$ Where a < c < b
(iv) $\int_{0}^{a}f(x)dx=\int_{0}^{a}f({\color{Blue} a-x})dx$
(v) $\int_{0}^{a}f(x)dx=\int_{0}^{a/2}f(x)dx+\int_{0}^{a/2}f({\color{Blue} a-x})dx$
(vi) $\int_{a}^{b}f(x)dx=\int_{a}^{b}f({\color{Blue} a+b-x})dx$
(vii)

(viii)

(ix)

(x) $\int_{{\color{Blue} mT}}^{{\color{Blue} nT}}f(x)dx=(n-m)\int_{0}^{T}f(x)dx$ f(x) is periodic with period T

Note: $If \ {\color{Blue} f(x)\geq g(x)} \ on \ [a,b] \ then \ {\color{Blue} \int_{a}^{b}f(x)dx\geq \int_{a}^{b}g(x)dx}$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics

Definite Integral

Definition:

Properties:

Definite integrals to the determination of areas involving simple curves

Example:

No comments:

Post a Comment