Integral Calculus Question 218
Question: If $ f(x) $ and $ \phi (x) $ are continuous functions on the interval $ [0,4] $ satisfying $ f(x)=f(4-x) $ , $ \phi (x)+\phi (4-x)=3 $ and $ \int\limits_0^{4}{f(x)dx=2,} $ then $ \int\limits_0^{4}{f(x)\phi (x)dx} $
Options:
A) 3
B) 6
C) 2
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \int\limits_0^{4}{f(x)\phi (x)dx=\int\limits_0^{4}{f(4-x)\phi (4-x)dx}} $ $ =\int\limits_0^{4}{f(x).(3-\phi (x))dx} $ $ [ \begin{aligned} & \because f(x)=f(4-x) \\ & and\phi (x)+\phi (4-x)=3 \\ \end{aligned} ] $ $ =3\int\limits_0^{4}{f(x)dx-I\Rightarrow 2I=3.2\therefore I=3} $
BETA
