What is the surface area of the solid created by revolving #f(x) = xe^-x-e^(x) , x in [1,3]# around the x axis?
The surface area is approximately 105.75.
which can be obtained by integration by parts, so
To make this physically meaningful, we would just take the absolute value of this and say that
And so we have
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To find the surface area of the solid created by revolving the function ( f(x) = xe^{-x} - e^x ) over the interval ( x ) in ([1,3]) around the x-axis, you can use the formula for the surface area of a solid of revolution:
[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
where ( y ) is the function being revolved, ( a ) and ( b ) are the interval endpoints, and ( \frac{dy}{dx} ) represents the derivative of ( y ) with respect to ( x ).
First, find the derivative of ( f(x) ):
[ f'(x) = e^{-x} - xe^{-x} - e^x ]
Then, calculate ( \left(\frac{dy}{dx}\right)^2 ):
[ \left(\frac{dy}{dx}\right)^2 = \left(e^{-x} - xe^{-x} - e^x\right)^2 ]
Now, substitute ( f(x) ), ( f'(x) ), and the interval ([1,3]) into the surface area formula:
[ S = \int_{1}^{3} 2\pi \left(xe^{-x} - e^x\right) \sqrt{1 + \left(e^{-x} - xe^{-x} - e^x\right)^2} , dx ]
This integral will give you the surface area of the solid created by revolving ( f(x) ) around the x-axis over the interval ([1,3]). You can evaluate this integral using numerical methods or software if an exact solution is not feasible.
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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.
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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