a Each value of x between 0 and \u03c0 gives a vertical strip of thickness dx that

# A each value of x between 0 and π gives a vertical

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(a) Each value of x between 0 and π gives a vertical strip, of thickness dx , that generates a disk when rotated about the x -axis. Thus we use the formula for volume of rotation via the disk method, V = Z π 0 π R 2 dx In our case, the radius of the disk is given by the distance from ( x , sin x ) to ( x , 0) for each x , and is therefore given by R = sin x . Hence V = Z π 0 π (sin x ) 2 dx We can compute this integral by applying the identity sin 2 ( x ) = 1 2 (1 - cos(2 x )) (see Problem 1, part (2)): V = π Z π 0 sin 2 ( x ) dx = π x 2 - sin(2 x ) 4 ! x = π x = 0 = π π 2 - sin(2 π ) 4 - 0 2 + sin(0) 4 ! = π π 2 - 0 - 0 + 0 = π 2 2 20
(b) Since we are now rotating about the (vertical) line x = - 2, our vertical strips no longer generate disks, but instead generate cylindrical shells. The formula for the volume of rotation via the cylindrical shell method is V = Z π 0 2 π · R · hdx Here R is the radius of rotation, given by the distance from x to the line of rotation at - 2; this distance is R = x + 2. The height h is the height of the vertical strip, equal to sin x as before in part (a). Thus V = Z π 0 2 π ( x + 2) sin( x ) dx = 2 π Z π 0 x sin( x ) dx + 2 π Z π 0 2 sin( x ) dx = 2 π Z π 0 x sin( x ) dx - 4 π cos( x ) | x = π x = 0 = 2 π Z π 0 x sin( x ) dx - (4 π · ( - 1) - 4 π · (1)) = 2 π Z π 0 x sin( x ) dx + 8 π The integral R π 0 x sin( x ) dx can be evaluated by an integration by parts, setting u = x and dv = sin( x ) dx , whereby du = dx and v = - cos( x ). This gives Z π 0 x sin( x ) dx = - x cos( x ) | x = π x = 0 - Z π 0 ( - cos( x )) dx = - π cos( π ) - 0 + Z π 0 cos( x ) dx = - π · ( - 1) + sin( x ) | x = π x = 0 = π + (0 - 0) = π Putting all this together, we have V = 2 π Z π 0 x sin( x ) dx + 8 π = 2 π ( π ) + 8 π = 2 π 2 + 8 π 2. As in part (1), we will use the disk / washer method for part (a) and the cylindrical shell method for part (b); although the function y = x - 5 is more easily inverted than the function in part (a), and we could exchange disk / washer and cylindrical shell methods by considering x = y - 1 / 5 , it seems easier to integrate over x since we are given the information about the region in terms of y as a function of x . 21
(a) We have as above vertical strips stretching from ( x , x - 5 ) to ( x , 0) for each x in [0 , ). However, unlike part (a), the axis of rotation is now y = - 2, not the x -axis; therefore the region under consideration does NOT border the axis of rotation, and we have to use the washer method V = Z 1 ( π R 2 - π r 2 ) dx Here R is the outer radius of rotation, given by the distance of the graph of the function y = x - 5 from the line y = - 2; so R = x - 5 + 2. The inner radius r is the distance from the inner curve y = 0 to the line y = - 2, which is r = 2 for all x . Therefore V = π Z 1 (( x - 5 + 2) 2 - 2 2 ) dx = π Z 1 ( x - 10 + 2 x - 5 + 4 - 4) dx = π Z 1 ( x - 10 + 2 x - 5 ) dx = π x - 9 - 9 + 2 x - 4 - 4 ! x = x = 1 = π (0 + 0 - ( - 1 / 9 - 1 / 4)) = π (1 / 9 + 1 / 4) = 13 π 36 (b) Once again we use the cylindrical shells formula V = Z 1 2 π · R · hdx where R is the radius of rotation, given by the distance from x to the y -axis; therefore R = x . The height is given by the height of the vertical strip which, as in part (a), is given by h = x - 5 .