Let \( R \) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \( R \) is revolved about the \( y \)-axis. \( y=5-x, y=0, x=1 \), and \( x=4 \) Set up the integral that gives the volume of the solid using the shell method. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. B.

\( \frac{d}{dx}(y^{3}\sqrt {x^{2}+y})=\frac{d}{dx}(x) \)

Suppose the interval \( [-5,-3] \) is partitioned into \( n=4 \) subintervals. What is the subinterval length \( \Delta x \) ? List the grid points \( x_{0}, x_{1}, x_{2}, x_{3}, x_{4} \) Which points are used for the left right, and midpoint Riemann sums? \( \Delta x=\square \)

2. \( \int_{2}^{2}\left(5 x^{2}+3 x+5\right) \cdot d x=\left({ }_{2}^{3}\left(5 x^{2}+8 x+5\right) d x\right. \) 3. \( \int_{2}^{3}\left(x^{3}+9\right) \cdot d x= \)

Use the price-demand equation to find the values of \( p \) for which demand is elastic and the values for which demand is inelastic. Assume that price and demand are both positive. \[ x=f(p)=2704-4 p^{2} \] The values of \( p \) for which demand is elastic are \( \square \) . (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) The values of \( p \) for which demand is inelastic are \( \square \) \( \square \). (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Las cuatro esquinas de la caja cuyos lados son mestán representadas por \( X \); se doblarán los bordes para formar una tapa de la caja. Determina la longitud de \( x \) para que el volumen sea máximo.