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How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=x^2#, y=2-x x=0 revolved about the y-axis?

Answer 1

Isabella Ackerman

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Answer 2

Ezra Adams

To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by (y=x^2), (y=2-x), and (x=0) revolved about the y-axis, follow these steps:

Determine the limits of integration by finding the points of intersection between the curves (y=x^2) and (y=2-x).
Set up the integral to find the volume using the formula for cylindrical shells: [V = 2\pi \int_a^b x \cdot h(x) , dx], where (h(x)) represents the height of the shell at each value of (x), and (a) and (b) are the limits of integration.
Calculate (h(x)) by finding the difference between the upper and lower functions at each (x) value.
Evaluate the integral using the determined limits of integration and the calculated (h(x)) function.

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Answer from HIX Tutor

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.