# What is the surface area of the solid created by revolving #f(x)=sqrt(x^3)# for #x in [1,2]# around the x-axis?

24.93

Surface area of solid of revolution about x axis is given by the formula

This integration is too long to write it here. hence use integral calculator to solve the integral

= 24.93

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To find the surface area ( S ) of the solid created by revolving ( f(x) = \sqrt{x^3} ) for ( x ) in ([1,2]) around the x-axis, we can use the formula for the surface area of a solid of revolution:

[ S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(f'(x)\right)^2} , dx ]

First, we need to find ( f'(x) ):

[ f(x) = \sqrt{x^3} ] [ f'(x) = \frac{d}{dx} \sqrt{x^3} ] [ f'(x) = \frac{1}{2\sqrt{x^3}} \cdot 3x^2 ] [ f'(x) = \frac{3x^2}{2\sqrt{x^3}} ]

Now, plug ( f(x) ) and ( f'(x) ) into the surface area formula:

[ S = 2\pi \int_{1}^{2} \sqrt{x^3} \sqrt{1 + \left(\frac{3x^2}{2\sqrt{x^3}}\right)^2} , dx ]

[ S = 2\pi \int_{1}^{2} \sqrt{x^3} \sqrt{1 + \frac{9x^4}{4x^3}} , dx ]

[ S = 2\pi \int_{1}^{2} \sqrt{x^3} \sqrt{1 + \frac{9x}{4}} , dx ]

Now, compute the integral:

[ S = 2\pi \int_{1}^{2} \sqrt{x^3} \sqrt{1 + \frac{9x}{4}} , dx ]

This integral is not straightforward to compute by hand due to the square root terms. You would typically use integration techniques such as substitution or numerical methods to evaluate this integral.

After evaluating the integral, you will get the surface area ( S ) of the solid formed by revolving ( f(x) = \sqrt{x^3} ) for ( x ) in ([1,2]) around the x-axis.

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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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