# How do you find the volume of the solid obtained by rotating the region bounded by the curves #x=0# and #Y=4-x^2# and #y=3x# rotated around the y-axis?

The cross-sections (head on views of vertical slices) for this solid of revolution will be "washers" (the more official name is annuli). The areas of the cross-sections will be given by

Based on a picture of the region you are revolving around the

Therefore, the cross-sectional area, as a function of

The volume of the resulting solid is given as a definite integral of

Hence

Here's a picture of the 2-dimensional region that gets revolved around the

And here's a picture of the 3-dimensional solid (with a different scale on the

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The volume of the solid obtained by rotating the region bounded by the curves (x = 0), (y = 4 - x^2), and (y = 3x) around the y-axis can be found using the method of cylindrical shells. The formula for the volume of the solid obtained by rotating a region around the y-axis using cylindrical shells is:

[ V = 2\pi \int_a^b x \cdot h(x) , dx ]

where (a) and (b) are the x-values of the intersection points of the curves, and (h(x)) is the height of the shell at the x-value (x).

In this case, the intersection points of the curves are (x = 0) and (y = 4 - x^2), so we need to find the x-values where (y = 4 - x^2) intersects with (y = 3x).

Setting (4 - x^2 = 3x) and solving for (x) gives (x^2 + 3x - 4 = 0), which factors to ((x + 4)(x - 1) = 0). So, the intersection points are (x = -4) and (x = 1).

The height of the shell (h(x)) is the difference between the y-values of the curves at the x-value (x), which is (h(x) = 4 - x^2 - 3x).

Now, we can calculate the volume using the formula:

[ V = 2\pi \int_{-4}^1 x \cdot (4 - x^2 - 3x) , dx ]

[ V = 2\pi \int_{-4}^1 (4x - x^3 - 3x^2) , dx ]

[ V = 2\pi \left[ 2x^2 - \frac{x^4}{4} - x^3 \right]_{-4}^1 ]

[ V = 2\pi \left[ (2(1)^2 - \frac{(1)^4}{4} - (1)^3) - (2(-4)^2 - \frac{(-4)^4}{4} - (-4)^3) \right] ]

[ V = 2\pi \left[ (2 - \frac{1}{4} - 1) - (32 - 64 - (-64)) \right] ]

[ V = 2\pi \left[ \frac{7}{4} - (-32) \right] ]

[ V = 2\pi \left[ \frac{7}{4} + 32 \right] ]

[ V = 2\pi \cdot \frac{135}{4} ]

[ V = \frac{270\pi}{4} ]

[ V = \frac{135\pi}{2} ]

So, the volume of the solid obtained by rotating the region around the y-axis is ( \frac{135\pi}{2} ).

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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.

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