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=1#, #y=x^2#, and #x=0# rotated about the line #y=2#?
The integral for the volume is
Integrating we get
Evaluating we get
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To 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=1), (y=x^2), and (x=0) rotated about the line (y=2), we first need to find the limits of integration and the expression for the radius and height of the cylindrical shells.
The limits of integration will be determined by the intersection points of the curves (y=1) and (y=x^2), which occur at (x=1) and (x=-1).
The expression for the radius of the cylindrical shells will be the distance from the axis of rotation ((y=2)) to the curve (y=x^2), which is (2 - x^2).
The expression for the height of the cylindrical shells will be the difference between the upper and lower (y)-values of the region, which is (1 - x^2).
Thus, the integral to find the volume of the solid is:
[ V = 2\pi \int_{-1}^{1} (2 - x^2)(1 - x^2) , dx ]
Solving this integral will give the volume of the solid generated by the rotation.
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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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- What is the volume of the solid produced by revolving #f(x)=sec, x in [pi/8,pi/3] #around the x-axis?
- What is the surface area of the solid created by revolving #f(x) =2x+5 , x in [1,2]# around the x axis?
- What is the arc length of #f(x)= sqrt(5x+1) # on #x in [0,2]#?

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