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 \(x=3\).

5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin ^{2} \theta \). En déduire la monotonie de \( r \). (c) Etudier les variations de \( t \mapsto r(t) \exp (-\arctan (t)) \) et en déduire que \( 0 \leq r(t) \leq r(0) \exp (\operatorname{arct} \) que \( r \) a une limite strictement positive en \( +\infty \).

\( \left. \begin{array} { l } { M = x y \cos x y + \sin x y } \\ { \frac { \partial M } { \partial y } = ? } \end{array} \right. \)

9/16/24, 1:33 PM 8. Use a change of variables to evaluate the definite integral. \( \int_{0}^{2} x \sqrt{4-x^{2}} d x \) \( \int_{0}^{2} x \sqrt{4-x^{2}} d x= \)

\( \lim _ { n i n } \frac { ( n + 1 ) ^ { 3 } - ( n + 2 ) ^ { 2 } } { ( n - 2 ) ^ { 3 } - ( n + 3 ) ^ { 3 } } \)

9/16/24, 1:33 PM 4. Use the definition of the definite integral to evaluate \( \int_{2}^{6}\left(x^{2}-4\right) d x \). \( \int_{2}^{6}\left(x^{2}-4\right) d x=\frac{160}{3} \) (Simplify your answer.)