Question

Consider the plane region P consisting of all points in the first quadrant lying above the x-axis and below the graph of y = 4 – x^2; it is bounded on the left by the y-axis, as shown in the figure below. A solid S is formed by revolving P around the y-axis. Using an appropriate choice of either horizontal or vertical slices, the slices will be circular washers of area A(t) = ?(R(t)^2 – r(t)^2) where t is either the variable x or the variable y (as appropriate), R(t) is the outer radius of the washer obtained by slicing at t and r(t) is the inner radius of the washer obtained by slicing at t (which could be 0 if the washer does not have a hole and is thus actually a disk). Thus the integral ?[a to b] A(t) dt for the volume of S can be written as ?[a to b] ?(R(t)^2 – r(t)^2) dt for appropriate values of a and b. (a) Are the slices horizontal (i.e., perpendicular to the y-axis) or vertical (i.e., perpendicular to the x-axis)? Enter either the word HORIZONTAL or the word VERTICAL. (b) What should the variable be, x or y? (c) What is the value of the lower limit of integration, a? (d) What is the value of the upper limit of integration, b? (e) What is the inner radius r of each slice, as a function of the variable? (f) What is the outer radius R of each slice, as a function of the variable?

          Consider the plane region P consisting of all points in the first quadrant lying above the x-axis and below the graph of y = 4 – x^2; it is bounded on the left by the y-axis, as shown in the figure below. A solid S is formed by revolving P around the y-axis. Using an appropriate choice of either horizontal or vertical slices, the slices will be circular washers of area A(t) = ?(R(t)^2 – r(t)^2) where t is either the variable x or the variable y (as appropriate), R(t) is the outer radius of the washer obtained by slicing at t and r(t) is the inner radius of the washer obtained by slicing at t (which could be 0 if the washer does not have a hole and is thus actually a disk). Thus the integral ?[a to b] A(t) dt for the volume of S can be written as ?[a to b] ?(R(t)^2 – r(t)^2) dt for appropriate values of a and b.

(a) Are the slices horizontal (i.e., perpendicular to the y-axis) or vertical (i.e., perpendicular to the x-axis)? Enter either the word HORIZONTAL or the word VERTICAL.
(b) What should the variable be, x or y?
(c) What is the value of the lower limit of integration, a?
(d) What is the value of the upper limit of integration, b?
(e) What is the inner radius r of each slice, as a function of the variable?
(f) What is the outer radius R of each slice, as a function of the variable?

Consider the plane region P consisting of all points in the first quadrant lying above the x-axis and below the graph of y = 4 – x^2; it is bounded on the left by the y-axis, as shown in the figure below. A solid S is formed by revolving P around the y-axis. Using an appropriate choice of either horizontal or vertical slices, the slices will be circular washers of area A(t) = ?(R(t)^2 – r(t)^2) where t is either the variable x or the variable y (as appropriate), R(t) is the outer radius of the washer obtained by slicing at t and r(t) is the inner radius of the washer obtained by slicing at t (which could be 0 if the washer does not have a hole and is thus actually a disk). Thus the integral ?[a to b] A(t) dt for the volume of S can be written as ?[a to b] ?(R(t)^2 – r(t)^2) dt for appropriate values of a and b.

(a) Are the slices horizontal (i.e., perpendicular to the y-axis) or vertical (i.e., perpendicular to the x-axis)? Enter either the word HORIZONTAL or the word VERTICAL.
(b) What should the variable be, x or y?
(c) What is the value of the lower limit of integration, a?
(d) What is the value of the upper limit of integration, b?
(e) What is the inner radius r of each slice, as a function of the variable?
(f) What is the outer radius R of each slice, as a function of the variable?

Added by Kathleen O.

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