The region under the curves #y=1/sqrtx, 1<=x<=2# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?

To sketch the region under the curve ( y = \frac{1}{\sqrt{x}} ) for ( 1 \leq x \leq 2 ) and find the volumes of the two solids of revolution when rotated about the x-axis:

1. Sketching the Region:

   • Plot the curve ( y = \frac{1}{\sqrt{x}} ) on the coordinate plane for ( 1 \leq x \leq 2 ).
   • Identify the region bounded by the curve and the x-axis between the given limits.

2. Finding Volumes of Solids of Revolution:

   • Split the region into two parts at the point where the curve intersects the x-axis (i.e., where ( y = 0 )).
   • For each part:
     • Determine the volume of the solid of revolution using the disk method.
     • Integrate the formula ( V = \pi \int_{a}^{b} [f(x)]^2 , dx ), where ( f(x) ) represents the function, and ( a ) and ( b ) represent the limits of integration.

3. Volume of First Solid:

   • For ( 1 \leq x \leq 2 ), the curve is above the x-axis.
   • Find the volume of revolution by integrating ( V_1 = \pi \int_{1}^{2} \left(\frac{1}{\sqrt{x}}\right)^2 , dx ).

4. Volume of Second Solid:

   • For ( 0 \leq x \leq 1 ), the curve is below the x-axis.
   • Find the volume of revolution by integrating ( V_2 = \pi \int_{0}^{1} \left(\frac{-1}{\sqrt{x}}\right)^2 , dx ).

5. Calculate the Values:

   • Evaluate the integrals to find the volumes ( V_1 ) and ( V_2 ).

6. Final Step:

   • Add the volumes of both solids to get the total volume of revolution about the x-axis.

This process will yield the volumes of the two solids of revolution formed by rotating the given region about the x-axis.