Here we consider various solids of revolution whose axis is the diagonal line y = x....

 

Transcribed Image Text:

Here we consider various solids of revolution whose axis is the diagonal line y = x. The goal is to adapt the "pile of thin disks" For each real constant m 1, the curve y = mx + (1 -m)r is a parabola that crosses the line y = x at the points (0,0) and (1,1). Let R(m) denote the finite region between the parabola and the line. Then let S(m) denote the solid generated by rotating R(m) around the line y = x. We are interested in the volume of the solid S(m): call this volume V(m). (a) Using the same set of axes, draw the line y = x and several of the parabolas y = mx + (1 - m)x. Your sketch should show enough parabolas to communicate all the different types of possible shapes. (b) Suppose m = -6. With reference to a suitable sketch, explain why the "pile of thin disks" idea cannot be used directly to find the volume V(-6). Then determine the largest closed interval [mo, m] of m-values for which this idea can be used directly. (c) Suppose m = 0. Let r(x) denote the radius of the disk of revolution whose centre has coordinates (x,x). Find a formula for r(r), valid for each number z in [0, 1]. Check that the values for r(0) and r(1) are compatible with the geometry of the situation. (d) Repeat part (c), but use m = 2. (e) Extend your reasoning in parts (c)-(d) to produce a formula for r(x) that involves m, and holds for each m (except m = 1) in the interval [mo, m] found in part (b). Note: Your formula should reproduce your earlier findings when m is replaced with either 0 or 2. (f) Suppose m / 1 lies in the interval [mo, m] found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume V(m), in terms of the function r(r) found earlier. r(x) dx. of Hint: The answer is not quite (g) Find the exact volume V(mo), where mo is the smallest number in the interval found in part (b). (h) Find the exact volume V(0). Here we consider various solids of revolution whose axis is the diagonal line y = x. The goal is to adapt the "pile of thin disks" For each real constant m 1, the curve y = mx + (1 -m)r is a parabola that crosses the line y = x at the points (0,0) and (1,1). Let R(m) denote the finite region between the parabola and the line. Then let S(m) denote the solid generated by rotating R(m) around the line y = x. We are interested in the volume of the solid S(m): call this volume V(m). (a) Using the same set of axes, draw the line y = x and several of the parabolas y = mx + (1 - m)x. Your sketch should show enough parabolas to communicate all the different types of possible shapes. (b) Suppose m = -6. With reference to a suitable sketch, explain why the "pile of thin disks" idea cannot be used directly to find the volume V(-6). Then determine the largest closed interval [mo, m] of m-values for which this idea can be used directly. (c) Suppose m = 0. Let r(x) denote the radius of the disk of revolution whose centre has coordinates (x,x). Find a formula for r(r), valid for each number z in [0, 1]. Check that the values for r(0) and r(1) are compatible with the geometry of the situation. (d) Repeat part (c), but use m = 2. (e) Extend your reasoning in parts (c)-(d) to produce a formula for r(x) that involves m, and holds for each m (except m = 1) in the interval [mo, m] found in part (b). Note: Your formula should reproduce your earlier findings when m is replaced with either 0 or 2. (f) Suppose m / 1 lies in the interval [mo, m] found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume V(m), in terms of the function r(r) found earlier. r(x) dx. of Hint: The answer is not quite (g) Find the exact volume V(mo), where mo is the smallest number in the interval found in part (b). (h) Find the exact volume V(0).

Answer rating: 100% (QA)

Final answer To visualize and determine the volume of solids ... View the full answer