1. Let S be the solid of revolution obtained by rotation around the line x=3 of the region R of the plane xy delimited by the curves x=4 , x=5 , y=5/x and y=1/x . We want to calculate the volume of S by the method of cylinders.

a) Let x be a real number in the interval [4,5] . We consider the thin cylinder obtained by rotation around the line x=3 of the portion of the region R between the abscissas x and x+x . What are the radius r of this cylinder and its height h? Give their formulas as a function of x.

r=

h=

b) The volume of this cylinder is approximately A(x)x . What is the function A(x)? (without x)

A(x)=

c) Determine approximately the volume of S to two decimal places. Responnse:

2.Let S be the flat-bottomed solid whose base is the region of the xy plane bounded by the curves y=e^x , y=3 , x=0 and x=2 , and whose sections perpendicular to the x-axis are squares whose base lies in the xy-plane.

a) Determine the area A(x) of the section of S by the plane of the abscissa points x .

A(x)= (exact formula)

b) Determine the volume of S to within 0.01. Answer:

3. Let S be the flat-bottomed solid whose base is the region of the xy plane bounded by the curves y=e^x , y=2 , x=0 and x=1 , and whose sections perpendicular to the x-axis are semicircles whose diameter lies in the xy plane.

a) Determine the area A(x) of the section of S by the plane of the abscissa points x .

A(x)= (exact formula)

b) Determine the volume of S to within 0.01. Answer: