15- 20 Use the method of cylindrical shells to find the volume of the solid obtained by rotating thg region bounded by the given curves about the \( x \)-axis. 15. \( x y=1, \quad x=0, \quad y=1, \quad y=3 \) 16. \( y=\sqrt{x}, \quad x=0, \quad y=2 \) 17. \( y=x^{3 / 2}, y=8, \quad x=0 \) 18. \( x=-3 y^{2}+12 y-9, \quad x=0 \) 19. \( x=1+(y-2)^{2}, \quad x=2 \)

Residues at Poles = Proof the Theorem Assume \( f(z) \) has the following form \[ f(z)=\frac{\phi(z)}{\left(z-z_{0}\right)^{m}} \] where \( \varphi(\mathrm{z}) \) is analytic and nonzero at \( \mathrm{z}_{0} \), then it has Taylor series representation

Calculate the area under the curve defined by the function \( f(x) = x^3 + 2x - 1 \) from \( x = 1 \) to \( x = 3 \).

3.2. \( \int \ln 3 x d x \)

احسب التكامل لـ \( \frac{4x^3 - 7x + 5}{2x + 1} dx \) من خلال تطبيق طريقة القسمة الطويلة.

(c) \( \int_{6}^{0}(2+5 x) e^{\frac{1}{3} x} d x \)