Question

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = 4x and y = x2/4 . Find V by slicing.

Answer 1

Answer:

Step-by-step explanation:

Consider the graphs of the $y = 4x$  and  $y = \frac{x^{2} }{4}$.

By equating the expressions, the intersection points of the graphs can be found and in this way delimit the area that will rotate around the Y axis.

$4x = \frac{x^{2} }{4} \\ x^{2} = 16x \\ x^{2} - 16x = 0 \\ x(x-16) = 0$ then $x=0$  o  $x=16$. Therefore the integration limits are:

$y = 4(0) = 0$  and  $y = 4(16) = 64$

The inverse functions are given by:

$x = 2 \sqrt{y}$  and  $x = \frac{y}{4}$. Then

The volume of the solid of revolution is given by:

$\int\limits^{64}_ {0} \, [2\sqrt{y} - \frac{y}{4}]^{2} dy = \int\limits^{64}_ {0} \, [4y - y^{3/2} + \frac{y^{2}}{16} ]\ dy = [2y^{2} - \frac{2}{5}y^{5/2} + \frac{y^{3}}{48} ]\limits^{64}_ {0} = 546.133 u^{2}$