The region under the curve #y=sqrt(1+x^2)# bounded by #0<=x<=1# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
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To sketch the region under the curve ( y = \sqrt{1 + x^2} ) bounded by ( 0 \leq x \leq 1 ) and find the volumes of the two solids of revolution:

Sketching the Region:
 Plot the curve ( y = \sqrt{1 + x^2} ) for ( 0 \leq x \leq 1 ).
 Draw the lines ( x = 0 ) and ( x = 1 ) to bound the region.
 Shade the area between the curve and the xaxis within the interval [0, 1].

Finding the Volume about the xaxis (Solid of Revolution):
 For rotation about the xaxis, the volume ( V_x ) can be calculated using the disk method: [ V_x = \pi \int_{a}^{b} [f(x)]^2 , dx ] where ( f(x) = \sqrt{1 + x^2} ), ( a = 0 ), and ( b = 1 ).
 Substitute the values into the formula and evaluate the integral to find ( V_x ).

Finding the Volume about the yaxis (Solid of Revolution):
 For rotation about the yaxis, the volume ( V_y ) can be calculated using the disk method: [ V_y = \pi \int_{c}^{d} x^2 , dy ] where ( x = \sqrt{y^2  1} ), ( c = 1 ), and ( d = \sqrt{2} ).
 Rewrite the integral in terms of ( y ) and ( x ), and evaluate it to find ( V_y ).

Once you have the volumes ( V_x ) and ( V_y ), you will have the volumes of the two solids of revolution.

Remember to perform any necessary simplifications and arithmetic calculations accurately to find the volumes of the 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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