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 #x= y^2# #x= y+2# rotated about the yaxis?
See explanation below.
To use shells when revoling about a vertical line, we need to take our representative slices vertically, so the thickness is
Here is a picture of the region to be revolved about the
The curves intersect at
The radii of our cylindrical shells will be
The thickness is
the height of each shell is the greater
(The
The greater value of
The lesser value of
From
To get that part of the volume, we need to evaluate:
From
To get that part of the volume, we need to evaluate:
The total volume is found by adding these two results, to get:
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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 yaxis, follow these steps:

Determine the limits of integration by finding the intersection points of the curves x = y^2 and x = y + 2.

Identify the variable of integration. Since we are revolving the region about the yaxis, the variable of integration will be y.

Determine the radius and height of the cylindrical shells. The radius (r) is the distance from the axis of rotation (yaxis) to the curve, which is given by the equation x = y^2. The height (h) of each shell is the difference between the two curves, given by (y + 2)  (y^2).

Set up the integral using the shell method formula: V = ∫[a, b] 2πrh dy, where a and b are the limits of integration.

Evaluate the integral.
Combining these steps, the integral to find the volume is:
V = ∫[a, b] 2πy(y + 2  y^2) dy
Once the integral is set up, you can evaluate it using standard techniques of integration.
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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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