# How do you find the volume of the region left of #y = sqrt(2x)# and below #y = 2# rotated about the y-axis?

So, you can use Guldino's formula for the volume of a solid of revolution:

So here we have

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To find the volume of the region left of (y = \sqrt{2x}) and below (y = 2) rotated about the y-axis, you can use the method of cylindrical shells.

- First, determine the limits of integration. Since the region is left of (y = \sqrt{2x}) and below (y = 2), you need to find the intersection point of these two curves. Set (y = \sqrt{2x}) equal to (y = 2) and solve for (x) to find the lower limit of integration.

(2 = \sqrt{2x})

Square both sides to solve for (x):

(4 = 2x)

(x = 2)

So, the lower limit of integration is (x = 2).

- The upper limit of integration is where (y = \sqrt{2x}) intersects the y-axis. Setting (x = 0) in (y = \sqrt{2x}) gives the upper limit of integration.

(y = \sqrt{2(0)})

(y = \sqrt{0})

(y = 0)

So, the upper limit of integration is (y = 0).

- Now, set up the integral using the formula for the volume of a cylindrical shell:

[V = 2\pi \int_{0}^{2} x \cdot (2 - \sqrt{2x}) , dx]

- Integrate with respect to (x) from (0) to (2):

[V = 2\pi \int_{0}^{2} (2x - x\sqrt{2x}) , dx]

- Solve the integral to find the volume.

[V = 2\pi \left[\frac{2x^2}{2} - \frac{2}{5}x^{5/2}\right] \Bigg|_{0}^{2}]

[V = 2\pi \left[x^2 - \frac{2}{5}x^{5/2}\right] \Bigg|_{0}^{2}]

[V = 2\pi \left[(2)^2 - \frac{2}{5}(2)^{5/2} - (0^2 - \frac{2}{5}(0)^{5/2})\right]]

[V = 2\pi \left[4 - \frac{8}{5}\right]]

[V = \frac{32\pi}{5}]

So, the volume of the region left of (y = \sqrt{2x}) and below (y = 2) rotated about the y-axis is (\frac{32\pi}{5}).

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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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