Chapter 7 curve. 3. x=y-y 2, x=0, about the y axis. 6. y=x, y= x,about y=1

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1 Chapter 7 curve Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line. Sketch the region, the solid, and a typical disk or washer.. y-/, =, =; about the. y= e, =, =; about the ais -ais 4. y=. =y ; about the -ais. 5. y=sec,=-, =; about the -ais. =y-y, =, about the y ais 6. y=,about y= Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line. Sketch the region, the solid, and a typical disk or washer. 7. y=, =-; about the ais. y= cos, =, 4 = about the ais =, y 8. y=-, = about the ais.. =y. and =y about the line y=-. 9. y= 5 about the =ais.. =-y, =+y - about the y ais. Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line using the shell method. Sketch the region, the solid, and the rectangular bo that would represent the shell.. y=, = about the y ais 4. y=, =4, =9 about the y ais 6. y=4, +y = 5 about the ais. 7. y=, =, = is revolved around the line =-. 5, y =, = about the - ais 8. y=, y =, = about the line y = Find the length of the following curves 9. y= from = to =. = (y + ) from y= to y= y= from = to =. y= (e + e ) from = to 6 =. y= from = to = = y + y from y= to 8 4 y=4

2 5. A solid has a circular base of radius 5. If every plane cross section perpendicular to the -ais is an equilateral triangle, then its volume is 6. A solid has a circular base of radius 5. If every plane cross section perpendicular to the -ais is a semi circle, then its volume is 7. A solid has a circular base of radius 5. If every plane cross section perpendicular to the -ais is an square, then its volume is 8. The base of a solid is the region enclosed by the ellipse 6 + y =. If all plane cross sections perpendicular to the -ais are semi-circles, then its volume is 9. The base of a solid is the region enclosed by the ellipse 6 + y =. If all plane cross sections perpendicular to the -ais are equilateral triangles, then its volume is. The base of a solid is the region enclosed by the ellipse 6 + y =. If all plane cross sections perpendicular to the -ais are squares, then its volume is. Find the length of the curve: y= from = to =9. Find the length of the curve:. Find the length of the curve: y = + from y= to y= 6 y lny = y from y= to y= 8 Set up the definite integral that gives the area of the region.. f ( ) = 4+, g ( ) = + +. f ( ) = ( ), g( ) = The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. 6 / ( ). 4 6 Approimation Determine which value best approimates the area of the region bounded by the graphs of f and g. (Make your selection on the basis of a sketch of the region and not by performing any calculations) 4.f()=-/, g()=- (a) (b) 6 (c) - (d) (e) 4 Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. 5. f ( ) = + +, g( ) = + 6. y=, y =, =, = 5 7. f ( y) = y, g(y) = y +

3 8. f ( y) = y, g(y) =, y = 6 y Use a graphing utility to graph the region bounded by the graphs of the functions and use the integration capabilities of the graphing utility to find the area of the region. 9. f ( ) = ( + ), g( ) =. 4 y =. f ( ) = /(+ ), g( ) = / Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y- ais. (Use the shell method).. y= 5. y = 4 4 Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y- ais. (Use the shell method). 4. y= / 8 indicated lines. 5. y=, y =, = (a) the y-ais (b) the -ais (c) the line y = 8 (d) the line = line y = y =, y =, = 7. y = sec, y =, line = = y, = 4 -ais. 9. y = 4. y= e, y =, =, = y-ais.. y = 9, =, = Use the integration capabilities of a graphing utility to approimate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -ais.. y= e, y =,, =, =. y = arctan(.), y =, =, = 5 4. Think about it (a) The region bounded by the parabola y= 4 and the -ais is revolved about the -ais. Find the volume of the resulting solid. (b) If the equation of the parabola in part (a) were changed to y= 4, would the volume of the solid generated be different? Eplain. 5. Use the disc method to verify that the volume of a right circular cone is / r h, where r is radius of the base and h is the height.

4 6. A cone with a base of radius r and height H is cut by a plane parallel to and h units above the base. Find the volume of the solid (frustum of a cone) below the plane. 7. A glass container can be modeled by revolving the graph of ,.5.95,.5 5 about the -ais, where and y are measured in centimeters. Use a graphing utility to graph the function and find the volume of the container. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-ais. 8. y= 9... y=, = y y, = 4 = 4, = sin, > y =, =, y = 4 Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the -ais.. y =, =, =, y = Use the shell method to find the volume of the solid generated by revolving the plane region about the indicated line.. y =, y = 4, about the line = 4 Use the disc or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. 4. y =, y =, = a) the -ais b) the y-ais c) the line = y = a a > (hypocycloid) a) the -ais b) the y-ais Use a graphing utility to graph the plane region bounded by the graphs of the equations and use the integration capabilities of the graphing utility to approimate the volume of the solid generated by revolving the region about the y-ais. 6. y = ( ) ( 6), y =, =, = Use integration to confirm your results in Eercise 9, where the region is bounded by the graphs of y=, y =, = 5 Find the arc length of the graph of the function over the indicated interval. 4. y = +, [,]

5 y = +,[,] Graph the function, highlighting the part indicated by the given interval, find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied thus far, and use the integration capabilities of a graphing utility to approimate the arc length. Function Interval 4. y= y = + y 45. = e y 46. = 6 y y 47. Approimate the arc length of the graph of the function over the interval [,4] in three ways. a) Use the Distance Formula to find the distance between the endpoints of the arc. b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when =, =, =, =, and =4. Find the sum of the four lengths. c)use the integration capabilities of a graphing utility to approimate the integral yielding the indicated arc length f ( ) = 48. Think about it Eplain why the two integrals are equal. e + d= + e d Use the integration capabilities of a graphing utility to verify that the integrals are equal.