Question

Some Properties of Definite Integrals.

limited Integrals.​

Answer 1

Answer:

Definition of Definite Integrals :- The definite integral is defined as an integral with two specified limits called the upper and the lower limit. The definite integral of a function generally represents the area under the curve from the lower bound value to the higher bound value.

SOME PROPERTIES OF DEFINITE INTEGRALS:-

PROPERTY 1 :-

p∫q f(a) da = p∫q f(t) dt

Property 2 :-

p∫q f(a) d(a) = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0

PROPERTY 3 :- p∫q f(a) d(a) = p∫r f(a) d(a) + r∫q f(a) d(a)

PROPERTY 4 :- p∫q f(a) d(a) = p∫q f( p + q – a) d(a)

PROPERTY 5 :- o∫p f(a) d(a) = o∫p f(p – a) d(a)

PROPERTY 6 :- ∫02p f(a)da = ∫0p f(a)da +∫0p f(2p-a)da…if f(2p-a) = f(a)

PROPERTY 7 :- 2 parts

2 parts :-

* ∫02 f(a)da = 2∫0a f(a) da … if f(2p-a) = f(a)

* ∫02 p f(a)da = 0 … if f(2p-a) = -f(a)

PROPERTY 8 :-

2 parts :-

* ∫-pp f(a)da = 2∫0p f(a) da … if f(-a) = f(a) or it’s an even function

*∫-ppf(a)da = 0 … if f(2p-a) = -f(a) or it’s an odd function

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Answer 2

Let a real function f(x) be defined and bounded on the interval [a,b]. The definite integral of the function f(x) over the interval [a,b] is defined as the limit of the integral sum (Riemann sums) as the maximum length of the subintervals approaches zero. ...