Use both the washer method and the shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \( y=x^{2}, y=4 \), and \( x=0 \) is revolved about the line \( y=-3 \). Set up the integral that gives the volume of the solid as a single integral if possible using the disk/washer method. Select the correct choice below and fill in any answer boxes within your choice. (Type exact answers.) A.

Suppose \( f \) is continuous on \( [1,5] \) and the only solutions of the equation \( f(x)= \) 6 are \( x=1 \) and \( x=4 \). If \( f(2)=8 \), explain why \( f(3)>6 \). Show that \( F(x)=|x| \) is continuous evervwhere.

Calcule \( \int \frac{x^{4} - 5x^{2} + 6}{x-2} \,dx \) aplicando o método de divisão longa.

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

By using Newton-Raphson Method, determine the root for \( x^{3}+2 x^{2}-1=0 \). Given \( x_{0}=0.5 \). Give the answer correct to four decimal places. Dengan mengomenan

\( \lim _ { x \rightarrow 0 } \frac { 3 ( x - \operatorname { cts } x ) } { x } \)