Question

Let ${\int }_0^{a}f\left(x\right)dx=\lambda$ and ${\int }_0^{a}f(2a-x)dx=\mu .$

Then ${\int }_0^{2a}f\left(x\right)dx$ is equal to?

• A. $\lambda +\mu$
• B. $\lambda -\mu$
• C. $2\lambda -\mu$
• D. $\lambda -2\mu$

Answer 1

The correct option is B $\lambda -\mu$

${\int }_0^{a}f\left(x\right)dx=\lambda$

$f\left(a\right)-f\left(0\right)=\lambda \to 1$

${\int }_0^{a}f(2a-x)dx=\mu$

$f\left(a\right)-f\left(2a\right)=\mu \to 2$

from 1 and 2

$f\left(a\right)-f\left(0\right)=\lambda$

$-f\left(2a\right)+f\left(a\right)=\mu$

+ - -

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$f\left(2a\right)-f\left(0\right)=\lambda -\mu$

${\int }_0^{2a}f\left(x\right)dx=\lambda -\mu$