Use the method of cylindrical shells to find the volume v of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 3 + y2, x = 0, y = 2, y = 3 v =

We have that  the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis is

From the question we are told

x = 3 + y2,

x = 0,

y = 2,

y = 3

Given a sketch of the Graph of x against y

Generally the equation for the volume   is mathematically given as

Therefore

the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis is

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Step-by-step explanation:

Let's give this a go here.  The volume formula for the shell method while rotating about a horizontal line is

where p(y) is the distance from the axis of rotation (the x-axis) to the center of the solid.  This is a positive distance and it is just y.

h(y) is the horizontal height of the function.  Our function starts at x = 0 and ends at the function itself, so h(y) = 3 + y^2.

In the shell method when rotating about a horizontal line, we need to use x = y equations, and y-intervals.  Setting up our integral then:

We can simplify this a bit by distributing the y into the parenthesis:

Integrating gives us

from 2 to 3

Using the First Fundamental Theorem of Calculus:

which simplifies down to

whats the