What is the volume obtained by rotating the region enclosed by #y=11-x#, #y=3x+7#, and #x=0# about the y-axis?
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To find the volume obtained by rotating the region enclosed by ( y = 11 - x ), ( y = 3x + 7 ), and ( x = 0 ) about the y-axis, you can use the method of cylindrical shells.
The formula for the volume using cylindrical shells is:
[ V = 2\pi \int_{a}^{b} x \cdot f(x) , dx ]
Where:
- ( f(x) ) is the height of the shell (the distance between the curves)
- ( a ) and ( b ) are the x-values where the region starts and ends.
First, find the points of intersection of the curves: [ 11 - x = 3x + 7 ] [ x = 2 ]
Now, integrate from ( x = 0 ) to ( x = 2 ):
[ V = 2\pi \int_{0}^{2} x \cdot [(11 - x) - (3x + 7)] , dx ]
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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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