How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=2x^2+5#, #y=x+3#, the yaxis, and the line #x=3# rotated about the xaxis?
See the explanation section, below.
Here is a graph of part of the region to be rotated about the
In order to use shells, we must take our representative slice parallel to the axis of rotation. In this case, that's the
That means we'll need to rewrite the curves as functions of
Every shell will have
As the graph shows, there are three separate integrals we need to do, because the calculation of
(Are we sure we want to use shells for this?)
From
So we need to integrate
From
So we need to integrate
From
So we need to integrate
Washers
To use washers take the slices perpendicular to the axis of rotation.
As x varies from
We need to integrate
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To use the shell method to set up and evaluate the integral for the volume of the solid generated by revolving the given plane region about the xaxis, you first need to determine the limits of integration and the radius and height of the shells.

Find the points of intersection of the curves (y = 2x^2 + 5) and (y = x + 3): (2x^2 + 5 = x + 3) (2x^2  x + 2 = 0) Solve for (x) to find the points of intersection.

The limits of integration will be from the xcoordinate of the leftmost point of intersection to the xcoordinate of the rightmost point of intersection.

For each shell, the radius is the xcoordinate of the shell, and the height is the difference between the ycoordinates of the two curves at that xcoordinate.

The volume of each shell is (2\pi rh), where (r) is the radius and (h) is the height. Integrate this expression over the limits of integration to find the total volume.
The integral to evaluate will look like:
[\int_{a}^{b} 2\pi x (y_{outer}  y_{inner}) , dx]
where (a) and (b) are the xcoordinates of the points of intersection, and (y_{outer}) and (y_{inner}) are the equations of the outer and inner curves, respectively.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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 How to you find the general solution of #dy/dx=sin2x#?
 What is the arc length of #f(x) = xxe^(x^2) # on #x in [ 2,4] #?
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