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How do you find the volume of a region that is bounded by #x=y^2-6y+10# and #x=5# and rotated about the y-axis?

Answer 1

Aurora Alvarado

Ok, have a look at this one:

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Answer 2

Christopher Allers

To find the volume of the region bounded by the curves (x = y^2 - 6y + 10) and (x = 5) rotated about the y-axis, you can use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:

[V = 2\pi \int_{a}^{b} x \cdot h(y) , dy]

where (h(y)) is the height of the shell and (a) and (b) are the limits of integration.

First, find the intersection points of the curves (x = y^2 - 6y + 10) and (x = 5). Setting them equal gives:

[y^2 - 6y + 10 = 5]

Solve this quadratic equation to find the bounds of integration.

Once you have the bounds, the height of the shell (h(y)) is the difference between the x-values of the curves at the given y-value.

Then, integrate (x \cdot h(y)) with respect to (y) from the lower bound to the upper bound.

This will give you the volume of the region.

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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.