GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 26 of 65

Marks: +1, -0

According to the Mean Value Theorem for a continuous function

f(x)

in the interval

[a, b]

, there exists a value

ξ

in this interval such that

∫↙{a}↖{b} f(x) d x=

Solution:

∫↙{a}↖{b} f(x) d x=b . f(b)-a . f(a)

....(1)
This implies that within interval

[a, b]

there exists a value

ξ

such that

f(ξ)=\;{b . f(b)-a ⋅ f(a)}/{b-a}

f(ξ)(b-a)=b . f(b)-a ⋅ f(a)

.....(2)
From (1)and (2),

∫↙{a}↖{b} f(x) d x=f(ξ)(b-a)

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