MCB4UW Handout 7.6. Comparison of the Disk/Washer and Shell Methods. V f x g x. V f y g y

Transcription

1 MCBUW Handout 7.6 Comparison of the Disk/Washer and Shell Methods Method Ais of Formula Notes aout the Revolution Representative Rectangle a Disk Method -ais V f d -ais a V g d Washer Method -ais a V f g d a -ais V f g d a Shell Method -ais V g d f is the length d is the width g is the length d is the width f is the top curve g d is the width f is the ottom curve is the right curve g d is the width is the left curve is the distance to the Ais of rotation g is the length d is the length -ais a V f d is the distance to the f Ais of rotation is the length d is the width It is important to note that the representative rectangle in the Disk and the Washer Methods are alwas going to e perpendicular to the ais of revolution. With the Shell Method, the representative rectangle will alwas e parallel to the ais of revolution.

2 987 Previous AP Sample Questions Let R e the region enclosed the graphs of 6 and a) Find the volume of the solid generated when region R is revolved around the -ais. ) Set up, ut do not integrate, an integral epression in terms of a single variale for the volume of the solid generated when region R is revolved aout the -ais Solutions in terms of Washer Method and Shell Method. a) Washer Method Points of intersection 6 6 6, 6 8 d 6 6 V d Shell Method as a function of 6 6 V d 6 5 d

3 ) Washer Method Shell Method V d BC V d 6 Let R e the region enclosed the graph of ln, and the line =, and the -ais. a) Find the area of region R. ) Find the volume of the solid generated revolving region R aout the -ais. c) Set up, ut do not integrate, an integral epression in terms of a single variale for the volume of the solid generated revolving region R aout line =. Solutions: a) A ln d ln ln or ln e ln A e d ln e ln Integration Parts u ln dv d du d v ln d ln d ln or We set f ln and g. D I ln + Thus

4 ln d d ln d ln ln ) Discs ln ln ln V d ln 6ln Integration Parts We set f ln and g. D I ln + ln Thus ln d ln ln d ln ln Shells V e d ln e e ln 6ln Integration Parts We set f and g e. D I + e e Thus e d e e d e e c) Discs ln V e d Shells V ln d

5 988 BC The ase of a solid S is the shaded region in the plane enclosed the -ais, the - ais, and the graph of sin, as shown. For each, the cross section of S perpendicular to the -ais at the point (, ) is an isosceles right triangle whose hpotenuse lies in the -plane. a) Find the area of the triangle as a function of ) Find the volume of S a) ) A h sin V a A d sin d sin sin cos sin d sin cos d

6 998 Let R e the region ounded the -ais, the graph of and the line =. a) Find the area of the region R ) Find the value h such that the vertical line =h divides the region R into two regions of equal area. c) Find the volume of the solid generated when R is revolved aout the -ais d) The vertical line =k divides the region R into two regions such that when these two regions are revolved aout the -ais, the generate solids with equal volumes. Find the value of k. a) ) 6 A d units h 8 d h 8 8 h h h.5 c) discs: V d d 8units d) k d k k 8.88

7 996 Let R e the region in the first quadrant under the graph for 9 a) Find the area of R ) If =k divides the region R into two regions of equal area, what is the value of k? c) Find the volume of the solid whose ase is the region R and whose cross sections cut planes perpendicular to the -ais are squares. a) ) 9 9 A d 9 k k d k k k 5 k c) 9 9 V d d 9 ln ln 9 ln.8

8 Let R e the shaded region in the first quadrant enclosed the graphs of cos, and the -ais, as shown in the figure. e, a) Find the area of region R ) Find the volume of the solid generated when the region R is revolved aout the - ais. c) The region R is the ase of a solid. For this solid, each cross section perpendicular to the -ais is a square. Find the volume of this solid. a) e cos A e cos d ) cos V e d.77 c) V A d a.99.6 e cos d