☛ Utiliser la relation de Chasles

Énoncé
Soit \(f\) la fonction définie sur \([0~;~3]\) par morceaux :  \(\begin{cases} f(x)=x \text{ si } x\in[0~;~1] \\ f(x)=1\text{ si }x\in ]1~;~2]\\ f(x)=-x+3 \text{ si } x\in ]2~;~3] \end{cases}\) .
Calculer  \(\displaystyle \int_0^ 3f(x)\text d x\) .

Solution

\(\begin{array}{rcl} \displaystyle \int_0^3f(x)\text dx &=&\displaystyle \int_0^1f(x)\text dx+\displaystyle \int_1^2f(x)\text dx+\displaystyle \int_2^3f(x)\text dx\\ \displaystyle \int_0^3f(x)\text dx &=&\displaystyle \int_0^1x \text dx+\displaystyle \int_1^2 1\text dx+\displaystyle \int_2^3 (-x+3)\text dx\\\displaystyle \int_0^3f(x)\text dx &=&\displaystyle \left[\dfrac12x^2\right]_0^1 +\bigg[x\bigg]_1^2+\displaystyle \left[-\dfrac12x^2+3x\right]_2^3\\\displaystyle \int_0^3f(x)\text dx &=&\dfrac12+2-1-\dfrac92+9-(-2+6)=2.\\\end{array}\)