# How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=sqrtx# and #y=x/3# rotated around the #x=-1#?

See the explanation section, below.

Here is a picture of the region (in blue) and the line

It shows a representative slice of thickness

The volume of the shell is

The height of the shell is the greater

We also need the point of intersection where

The integral we need to evaluate is:

Expand the integrand to get 4 terms in powers of

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To find the volume of the solid obtained by rotating the region bounded by the curves (y=\sqrt{x}) and (y=\frac{x}{3}) around the line (x=-1), you can use the method of cylindrical shells.

The limits of integration will be determined by the points where the two curves intersect. Setting ( \sqrt{x} = \frac{x}{3} ), you can solve for (x) to find the intersection point.

( \sqrt{x} = \frac{x}{3} )

( x = \frac{x^2}{9} )

( 9x = x^2 )

( x^2 - 9x = 0 )

( x(x - 9) = 0 )

(x = 0) or (x = 9)

So, the limits of integration will be (x = 0) to (x = 9).

The radius (r) of the cylindrical shell at (x) is (x + 1), because the axis of rotation is (x = -1). The height (h) of the cylindrical shell at (x) is the difference between the (y)-values of the upper and lower curves, which is (\sqrt{x} - \frac{x}{3}).

The volume (V) of each cylindrical shell is given by (V = 2\pi rh).

Therefore, the total volume is given by the integral:

[V = \int_{0}^{9} 2\pi(x + 1)\left(\sqrt{x} - \frac{x}{3}\right) dx]

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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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