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 y-axis, and the line #x=3# rotated about the x-axis?
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 x-axis, you first need to determine the limits of integration and the radius and height of the shells.
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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.
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The limits of integration will be from the x-coordinate of the leftmost point of intersection to the x-coordinate of the rightmost point of intersection.
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For each shell, the radius is the x-coordinate of the shell, and the height is the difference between the y-coordinates of the two curves at that x-coordinate.
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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 x-coordinates 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 one-sided 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 one-sided 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 one-sided 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 one-sided 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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