READ CAREFULLY: The region

R

is formed by

y=x^(2),y=0

, and

x=1

. Imagine rotating

R

around the

x

- axis to create an empty shell, and choosing

y

to be the independent variable for cylinders used to fill the shell and accumulate volume.\ NOTE: This is the SAME region

R

, rotated the SAME way, to create the SAME empty shell as the previous problems

26.-30

. The only difference is now using

y

as the independent variable which results in different cylinders that fill the shell. DRAW PICTURES.\ 31. Which of a)-e) best depicts the cylinders used to fill the shell in question?\ a)\ As

y

varies through

\\\\Delta y

-intervals, a new cylinder is always being generated in the current

\\\\Delta y

interval according to

dy

as it varies in the current interval.\ 32. What is the radius of the cylinder? If there is an inner and outer radius, this question is for the outer radius.\ a)

y

\ b)

\\\\sqrt()y

\ c) 1\ d)

1-\\\\sqrt()y

\ e)

y-1

\ 33. What is the height of the cylinder? (Hint: Draw a typical cylinder in a sketch of

R

and the empty shell. Is the cylinder height an

x

-coord,

y

-coord, large variation in

x

, or a large variation in

y

?)\ a)

y

\ b)

\\\\sqrt()y

\ c) 1\ d)

1-\\\\sqrt()y

\ e) None of these

31. - 35. READ CAREFULLY: The region R is formed by y=x2,y=0, and x=1. Imagine rotating R around the x - axis to create an empty shell, and choosing y to be the independent variable for cylinders used to fill the shell and accumulate volume. NOTE: This is the SAME region R, rotated the SAME way, to create the SAME empty shell as the previous problems 26 . -30 . The only difference is now using y as the independent variable which results in different cylinders that fill the shell. DRAW PICTURES. 31. Which of a)-e) best depicts the cylinders used to fill the shell in question? a) b) c) d) e) As y varies through y-intervals, a new cylinder is always being generated in the current y interval according to dy as it varies in the current interval. 32. What is the radius of the cylinder? If there is an inner and outer radius, this question is for the outer radius. a) y b) y c) 1 d) 1y e) y1 33. What is the height of the cylinder? (Hint: Draw a typical cylinder in a sketch of R and the empty shell. Is the cylinder height an x-coord, y-coord, large variation in x, or a large variation in y ?) a) y b) y c) 1 d) 1y e) None of these