How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=f(x)=3xx^2# and x axis revolved about the x=1?
The volume of the cylindrical shell is width x length x thickness (height)
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To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by (y = f(x) = 3x  x^2) and the xaxis, revolved about the line (x = 1), follow these steps:

Identify the region of interest, which in this case is the area under the curve (y = f(x) = 3x  x^2) bounded by the xaxis and the yaxis.

Determine the limits of integration. Since we're revolving about the line (x = 1), the limits of integration will be from the smallest xvalue of the region to the largest xvalue of the region. To find these limits, set (3x  x^2 = 0) and solve for x.
(3x  x^2 = 0)
(x(3  x) = 0)
(x = 0) and (x = 3)
So, the limits of integration are from (x = 0) to (x = 3).

Set up the integral for the volume using the formula for cylindrical shells:
(V = 2\pi \int_{a}^{b} x \cdot f(x) , dx)
where (a) and (b) are the lower and upper limits of integration respectively.

Substitute the function (f(x) = 3x  x^2) into the integral.
(V = 2\pi \int_{0}^{3} x \cdot (3x  x^2) , dx)

Simplify the expression and integrate.

Evaluate the integral to find the volume of the solid obtained by rotating the region bounded by (y = f(x) = 3x  x^2) and the xaxis, revolved about the line (x = 1).
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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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