The region under the curves #y=sqrt(e^x+1), 0<=x<=3# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
Please see below. (There is only one solid.)
To sketch
And at
Arithmetic approximation:
(
The derivative:
(For a better sketch investigate concavity. The graph is concave up.)
The region is shaded blue.
To go around the
The volume of a representative disk is
Since
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To sketch the region under the curve ( y = \sqrt{e^x + 1} ) for ( 0 \leq x \leq 3 ) and find the volumes of the two solids of revolution when this region is rotated about the x-axis, follow these steps:
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Sketch the curve ( y = \sqrt{e^x + 1} ) for ( 0 \leq x \leq 3 ). This will give you the upper boundary of the region.
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Since the lower boundary is the x-axis, the region is bounded by the curve ( y = \sqrt{e^x + 1} ) and the x-axis for ( 0 \leq x \leq 3 ).
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To find the volumes of the two solids of revolution, divide the region into two parts at the point where the curve intersects the x-axis. This point can be found by setting ( \sqrt{e^x + 1} = 0 ) and solving for x. However, since the square root of a positive number is always positive, the curve does not intersect the x-axis for ( 0 \leq x \leq 3 ). This means that the entire region is above the x-axis.
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The first solid of revolution is generated by rotating the region above the x-axis about the x-axis. To find its volume, use the formula ( V_1 = \pi \int_{0}^{3} [\sqrt{e^x + 1}]^2 , dx ).
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The second solid of revolution is the region below the x-axis (which in this case is empty) rotated about the x-axis. Its volume is given by ( V_2 = \pi \int_{0}^{3} [\sqrt{e^x + 1}]^2 , dx ).
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Evaluate the integrals to find the volumes of the two solids.
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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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