What is the volume of the solid produced by revolving #f(x)=xe^x(x/2)e^x, x in [2,7] #around the xaxis?
The volume of the solid produced by revolving around the xaxis the trapezoid of the function
The integral is solved iteratively by parts:
Finally:
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To find the volume of the solid produced by revolving the function ( f(x) = xe^x  \frac{x}{2}e^x ) on the interval ( x \in [2,7] ) around the xaxis, you can use the method of cylindrical shells:

Determine the radius of each cylindrical shell. Since we're revolving around the xaxis, the radius is the distance from the axis to the function, which is ( f(x) ).

Determine the height of each cylindrical shell. The height of each shell is the differential element along the xaxis, which is ( dx ).

Write the volume element ( dV ) for each cylindrical shell as ( dV = 2\pi x f(x) dx ), as the circumference of a cylindrical shell is ( 2\pi x ) and the height is ( f(x) ).

Integrate ( dV ) from ( x = 2 ) to ( x = 7 ) to find the total volume: [ V = \int_{2}^{7} 2\pi x f(x) , dx ]

Compute the definite integral to find the volume.

The resulting value will be the volume of the solid produced by revolving ( f(x) ) around the xaxis on the given interval.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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