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 xaxis, follow these steps:

Sketch the curve ( y = \sqrt{e^x + 1} ) for ( 0 \leq x \leq 3 ). This will give you the upper boundary of the region.

Since the lower boundary is the xaxis, the region is bounded by the curve ( y = \sqrt{e^x + 1} ) and the xaxis for ( 0 \leq x \leq 3 ).

To find the volumes of the two solids of revolution, divide the region into two parts at the point where the curve intersects the xaxis. 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 xaxis for ( 0 \leq x \leq 3 ). This means that the entire region is above the xaxis.

The first solid of revolution is generated by rotating the region above the xaxis about the xaxis. To find its volume, use the formula ( V_1 = \pi \int_{0}^{3} [\sqrt{e^x + 1}]^2 , dx ).

The second solid of revolution is the region below the xaxis (which in this case is empty) rotated about the xaxis. Its volume is given by ( V_2 = \pi \int_{0}^{3} [\sqrt{e^x + 1}]^2 , dx ).

Evaluate the integrals to find the volumes of the two solids.
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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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