If ( ) is approximated by a left sum using three inscribed rectangles of equal width on the x-axis, then the approximation is

Transcription

1 More Integration Page 1 Directions: Solve the following problems using the available space for scratchwork. Indicate your answers on the front page. Do not spend too much time on any one problem. Note: Let ln(x) denote the natural logarithm of x with base e If ( ) is approximated by a left sum using three inscribed rectangles of equal width on the x-axis, then the approximation is Page 1 of 21

2 More Integration Page ( ) For the values of a continuous function given in the table above, ( ) is approximated by a Riemann sum using the value at the midpoint of each of the three intervals of width 2. The approximation is Page 2 of 21

3 More Integration Page 3 03 The area between the line and the curve is 04 The area between and is Page 3 of 21

4 More Integration Page 4 05 The curves ( ) and ( ) shown in the figure above intersect at the point ( ) The area of the shaded region, bounded by these curves and the coordinate axes, is given by ( ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) Page 4 of 21

5 More Integration Page 5 06 If has the graph shown below, then ( ) ( ) ( ) ( ) ( ) ( ) Page 5 of 21

6 More Integration Page 6 07 The area enclosed by the curves ( ) and ( ) and the lines and is given by ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) Page 6 of 21

7 More Integration Page 7 08 If and the area of the region in the first quadrant under the graph of ( ) from 0 to is 0.1, then 09 The area of the region bounded by ( ) and ( ) for is Page 7 of 21

8 More Integration Page 8 10 The region in the first quadrant between the x-axis and the graph from to, is rotated about the line The volume of the resulting solid of revolution is given by ( ) ( ) ( ) ( ) ( ) 11 The curve between and is rotated around the y-axis. What volume is generated between it and the y-axis? Page 8 of 21

9 More Integration Page 9 12 The volume generated by revolving the region bounded by the y-axis and for around the y-axis is 13 What is the volume of the solid obtained by rotating the region between and around the x-axis? [( ) ( ) ] [( ) ( ) ] [( ) ( ) ] [( ) ( )] [( ) ( )] Page 9 of 21

10 ( ) AP Calculus BC More Integration Page The volume of the solid obtained by rotating about the x-axis the region in the first quadrant bounded by is ( ) ( ) ( ) 15 The region in the first quadrant between the x-axis from to and the graph, is rotated about the x-axis. The volume of the resulting solid of revolution is given by ( ) Page 10 of 21

11 More Integration Page The region in the first quadrant between the graphs of and is rotated around the x-axis. The volume of the resulting solid of revolution is 17 The region in the first quadrant bounded by the graphs of ( ) and the axes is rotated around the x-axis. The volume of the resulting solid of revolution is Page 11 of 21

12 More Integration Page The region in the first quadrant between the x-axis and the graph of is rotated around the y-axis. The volume of the resulting solid of revolution is given by ( ) ( ) ( ) ( ) ( ) 19 The base of a solid is the region enclosed by the graph of, the coordinate axes, and the line x = 3. If all place cross-sections perpendicular to the x-axis are squares, then its volume is ( ) Page 12 of 21

13 More Integration Page The roof and walls of a storage building are built in the shape modeled by the curve ( ) Each cross section cut perpendicular to the x-axis is a rectangle with a base of 50 feet and a height of y feet. In cubic feet, the volume of the building is approximately 21 The base of a solid is the region enclosed by the graph of and the y-axis. If all plane cross-sections perpendicular to the x-axis are semicircles with diameters parallel to the x-axis, then the volume is ) ) Page 13 of 21

14 More Integration Page If the length of a curve ( ) from to is given by then ( ) is 23 Which of the following definite integrals gives the length of between and ( ) ( ) ( ) ( ) ( ) Page 14 of 21

15 More Integration Page The length of the curve from ( ) to ( ) is If is a differentiable function such that the slope of at each point ( ( )) is then the length of the graph of between ( ( )) and ( ( )) is 2 Page 15 of 21

16 More Integration Page If ( ) is continuous and ( ) ( ( )) what is the length of the graph of ( ) from to ( ) ( ) ( ( )) ( ) ( ( )) 27 The length of the arc of from to is given by ( ) ( ) ( ) ( ) ( ) Page 16 of 21

17 More Integration Page The length of the arc joining ( ) and ( ) on the curve, is ( ) ( ) None of the above 29 What is the length of the arc of ( ) from to ( ) ( ) Page 17 of 21

18 More Integration Page Which set of inequalities is satisfied for the function ( ) on ( ) ( ) ( ) ( ) ( ) 31 The length of the arc given by, for is Page 18 of 21

19 More Integration Page Which of the following gives the surface area of the solid generated by revolving the arc from to about the x-axis? 33 The surface area generated by rotating about the x-axis the curve for is None of the above Page 19 of 21

20 More Integration Page The surface area generated by rotating about the x-axis the curve, for is 35 Which of the following gives the area of the surface generated by revolving about the x-axis the arc of from to Page 20 of 21

21 More Integration Page Which of the following gives the area of the surface generated by revolving about the x-axis the arc of from to ( ) ( ) ( ) Page 21 of 21