The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).

Calcula el área entre las curvas \( y = x^3 - 3x \) y \( y = x + 1 \) sobre el intervalo definido por sus intersecciones.

Application 3: Simplifier les expressions suivantes: \( \begin{array}{ll}A=\ln (\sqrt{3})+\ln 3-\ln 9 & ; \\ C=\ln \left(\frac{1}{3}\right)+\ln \left(\frac{3}{5}\right)+\ln \left(\frac{5}{7}\right)+\ln \left(\frac{7}{9}\right) & ;\end{array} \quad D=\ln (\sqrt[3]{2}-\ln 16 \)

\( \int _{0}^{\frac{\pi }{2}}\frac{1}{2+\cos (x)} d x \)

1) Sout \( x>0 \). En appliquant le thécorème des accroissement fines à la fonction \( \ln \) montrer que: \( \exists c \in] x, x+1\left[\frac{1}{c}=\ln \left(1+\frac{1}{x}\right)\right. \) 2) \( M q: \exists c \in] 0, x\left[\right. \) arctan \( (x)=\frac{x}{1+c^{2}} \) 3) En déduire que \( \forall x>0: \) \( \frac{1}{1+x}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x} \)

кцияның туындысын табыңыз: \( \left\{\begin{array}{l}x=t^{3}+3 t^{2}-5 \\ y=3 t^{2}+6 t-3\end{array}\right. \)