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#?

Answer 1

See the explanation section, below.

Here is a picture of the region (in blue) and the line #x=-1#.

It shows a representative slice of thickness #dx#. The slice is taken parallel to the axis of rotation, so when we rotate we'll get cylindrical shell.
The volume of the shell is #2pirhdx#. The radius, #r#, is shown by the dotted red horizontal line. Its length is #x+1#.
The height of the shell is the greater #y# value minus the lesser #y# value. That is: #h = sqrtx-x/3#

We also need the point of intersection where #x/3=sqrt3#. Which happens at #0# and at #9#

The integral we need to evaluate is:

#int_0^9 2 pi (x+1)(sqrtx-x/3) dx#

Expand the integrand to get 4 terms in powers of #x# which is straightforward to evaluate.

#2 pi int_0^9 (x+1)(sqrtx-x/3) dx = 207/5pi#

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Answer 2

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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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.

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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