Easy! But worth solving once

Calculus Level 3

Consider a function f ( x ) f(x) which is continuous for x [ 3 , 3 ] x\in [-3,3] .

If f ( x ) f(x) only attains(or gives output to) rational values in this domain and f ( 2 ) = 5 f(2)=5 then find:

3 3 f ( x ) d x \displaystyle \int_{-3}^{3}f(x)\,dx


The answer is 30.

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1 solution

Miraj Shah
Mar 22, 2016

Before solving this question let us understand one thing. Consider a number line and consider a rational point on it. We know that there are infinite irrational points lying just next to it on its either sides. If we understand this thing then the problem is just a cake walk!

Since the question says that the function f ( x ) f(x) only attains rational values and at the same time it is continuous , we can infer that the function is a constant function as even the slightest bend in the curve will mean that it does attain irrational values.

Now since its given that f ( 2 ) = 5 f(2) = 5 therefore we can conclude that f ( x ) = 5 f(x)=5 .

Therefore 3 3 f ( x ) d x = 5 [ x ] 3 3 = 30 \displaystyle \int_{-3}^{3} f(x)\,dx = 5[x]_{-3}^{3} = 30

Nice explanation bro

Santosh Narva - 5 years, 2 months ago

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