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=x^(1/2)#, #y=0#, and #x=4# rotated about the x axis?
This solid toberevolved looks like:
graph{(y  sqrtx)(y)sqrt(4(x0))/sqrt(4(x0)) <= 0 [5, 5, 0, 5]}
If you want to do it with the shell method, convert your functions to their inverses.
graph{(y  x^2)sqrt(4(y0))/sqrt(4(y0)) >= 0 [0, 8, 1, 5]}
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To use the shell method to find the volume of the solid generated by revolving the given plane region about the xaxis, follow these steps:

Identify the interval of integration for x. In this case, the interval is from x = 0 to x = 4, as given by the intersection points of y = x^(1/2) and the xaxis.

Determine the radius of each shell. Since we are revolving about the xaxis, the radius of each shell is the distance from the axis of rotation (xaxis) to the curve. Therefore, the radius is simply x.

Determine the height of each shell. The height of each shell is the difference between the upper and lower curves at each xvalue. In this case, the upper curve is y = x^(1/2) and the lower curve is y = 0. So, the height is simply y = x^(1/2).

Setup the integral to compute the volume using the shell method:
[V = \int_{0}^{4} 2\pi x (x^{1/2}) , dx]
 Evaluate the integral:
[V = 2\pi \int_{0}^{4} x^{3/2} , dx]
[V = 2\pi \left[\frac{2}{5}x^{5/2}\right]_{0}^{4}]
[V = 2\pi \left(\frac{2}{5} \cdot 4^{5/2}  \frac{2}{5} \cdot 0^{5/2}\right)]
[V = 2\pi \left(\frac{2}{5} \cdot 32  0\right)]
[V = \frac{64}{5}\pi]
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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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