Let R denote the finite region enclosed by the curves y = 1 and y= 2....

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Let R denote the finite region enclosed by the curves y = 1 and y= 2². Let S denote the solid obtained by rotating R about y-axis. We are interested the volume of the solid S: call this volume V. (a) Make a good sketch of the region R. (b) Imagine S as a stack of infinitesimally thin circular disks perpendicular to the y-axis. (This in- terpretation is used in several examples in section 1.6 of the CLP-2 text.) Writing dV for the infinitesimal volume of the disk at level y and thinking off as a "continuous sum" leads to the conceptual equation V = f dV. Interpreted this way, the volume of S may be written as V = [₁ _ dv = f* 1 (1) dy dV= for some function f(y). Find f(y). (c) Next, imagine S as a collection of infinitesimally thin cylindrical shells centred around the y-axis. (This interpretation is used in an example in section 1.6.1 of the CLP-2 textbook.) Writing dV for the infinitesimal volume of the specific shell passing through a particular point on the z-axis gives a new interpretation to the conceptual equation V=JdV. Now the volume of S may be written as v = ['_ av = ['₁90 dV= g(z) dz for some function g(z). Find g(z). (d) Show by direct calculation that both approaches above give the same result for the volume V. Let R denote the finite region enclosed by the curves y = 1 and y= 2². Let S denote the solid obtained by rotating R about y-axis. We are interested the volume of the solid S: call this volume V. (a) Make a good sketch of the region R. (b) Imagine S as a stack of infinitesimally thin circular disks perpendicular to the y-axis. (This in- terpretation is used in several examples in section 1.6 of the CLP-2 text.) Writing dV for the infinitesimal volume of the disk at level y and thinking off as a "continuous sum" leads to the conceptual equation V = f dV. Interpreted this way, the volume of S may be written as V = [₁ _ dv = f* 1 (1) dy dV= for some function f(y). Find f(y). (c) Next, imagine S as a collection of infinitesimally thin cylindrical shells centred around the y-axis. (This interpretation is used in an example in section 1.6.1 of the CLP-2 textbook.) Writing dV for the infinitesimal volume of the specific shell passing through a particular point on the z-axis gives a new interpretation to the conceptual equation V=JdV. Now the volume of S may be written as v = ['_ av = ['₁90 dV= g(z) dz for some function g(z). Find g(z). (d) Show by direct calculation that both approaches above give the same result for the volume V.