Integral exercise asked in csb.gov.lb (مباراة تعيين استاذ تعليم ثانوي 2004 ) the 2nd exercise

Exercise:

In the adjacent figure we have:

$\left( d \right)$ is a straight line of equation $x=10$ and

curve $\left( C' \right)$ is obtained from $\left( C \right)$ by

the translation vector $\vec{v}=3\vec{j}$

Let $\left( D \right)$ be the Domain limited by curves $\left( C \right)\,\,\And \,\left( C' \right)$

 and the line $\left( d \right)$ & $y-axis$

Calculate the Area of domain $\left( D \right)$

Solution:  we have $\left( C' \right)$ is obtained from $\left( C \right)$ by the translation vector $\vec{v}=3\vec{j}$

So the functions that represent the curves $\left( C \right)\,\And \,\left( C' \right)$ are the same

So if $\left( C \right)$ has function $f\left( x \right)$ then $\left( C' \right)$ has the function $f\left( x \right)+3$

Thus $A=\int_{0}^{10}{\left( {{f}_{\left( c' \right)}}-{{f}_{\left( c \right)}} \right)dx=\int_{0}^{10}{\left( f\left( x \right)+3-f\left( x \right) \right)dx}=3\int_{0}^{10}{dx=30\,\,uni{{t}^{2}}}}$