Find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^{2} \) and \( y = 4 \) about the line \( y = 5 \) using the washer method.

\( \frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 } \)

soit \( x \in \mathbb{R}^{+} \): On pose \[ A=\sqrt{x^{2}+1}-x \text { et } B=\sqrt{x^{2}+1}+x . \] 1. [1 pt] Montrer que : \( A>0 \) et déduire que : \( B>2 x \). 2. [1 pt] Calculer \( A B \) et déduire que : \( A<\frac{1}{2 x} \) pour \( x \neq 0 \). 3. [1 pt]Démontrer que pour tout \( x \neq 0: \quad x<\sqrt{x^{2}+1}<x+\frac{1}{2 x} \). 4. \( \left[1 \mathrm{pt}\right. \) ] Donner un encadrement d'amplitude \( \frac{1}{66} \) pour le nombre \( \frac{\sqrt{122}}{3} \). Exercice \( 4 ; 3.5 \) POINTS.

(a) Find \( \int \frac{x+2}{x^{2}+2 x-3} \mathrm{~d} x \) (b) Find \( \int 2 x \sin x \mathrm{~d} x \) (c) Using the substitution \( x=3 \sin \theta \), find the exact value of \( \int_{0}^{\frac{3}{2}} \sqrt{9-x^{2}} d x \) 9 (a) \( \frac{3}{4} \ln |x-1|+\frac{1}{4} \ln |x+3|+C \) or \( \frac{1}{2} \ln \left|x^{2}+2 x-3\right|+\frac{1}{4} \ln \left|\frac{x-1}{x+3}\right|+C \) (b) \( -2 x \cos x+2 \sin x+C \) (c) \( \frac{9}{8} \sqrt{3}+\frac{3}{4} \pi \)

9(a) Find \( \int \frac{x+2}{x^{2}+2 x-3} \mathrm{~d} x \) \( \begin{array}{ll}\text { (b) Find } \int 2 x \sin x \mathrm{~d} x\end{array} \) \( \begin{array}{ll}\text { (c) Using the substitution } x=3 \sin \theta \text {, find the exact value of } \int_{0}^{\frac{3}{2}} \sqrt{9-x^{2}} \mathrm{~d} x \\ 9 & \text { (a) } \frac{3}{4} \ln |x-1|+\frac{1}{4} \ln |x+3|+C \text { or } \frac{1}{2} \ln \left|x^{2}+2 x-3\right|+\frac{1}{4} \ln \left|\frac{x-1}{x+3}\right|+C \\ \text { (b) }-2 x \cos x+2 \sin x+C & \text { (c) } \frac{9}{8} \sqrt{3}+\frac{3}{4} \pi\end{array} \) \( \begin{array}{ll}\text { (b) }\end{array} \)

Evaluate the triple integral. \[ \iiint_{G} x y \sin (y z) d V \text {, } \] where \( G \) is the rectangular box defined by the inequalities \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)