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

Parametric Equations

Substitute in

When

Determining the Volume of a Solid of Revolution

For this Integration, You will use

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To find the surface area of the solid formed by revolving the curve ( f(t) = (3t-1, t^2-2t+2) ) around the x-axis from ( t = 2 ) to ( t = 3 ), you can use the formula for the surface area of revolution:

[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dt ]

Where:

- ( a ) and ( b ) are the limits of integration (in this case, ( a = 2 ) and ( b = 3 ))
- ( y ) represents the ( y )-coordinate of the curve ( f(t) )
- ( \frac{dy}{dx} ) represents the derivative of ( y ) with respect to ( x )

First, find the derivative ( \frac{dy}{dx} ) by differentiating the equation ( y = t^2-2t+2 ) with respect to ( t ). Then substitute into the formula and integrate over the given interval to find the surface area.

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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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