Volume by Disk/Washers - Classwork

Transcription

1 Volume by Disk/Washers - Classwork Example 1) Find the volume if the region enclosing y = x, y = 0, x = 3 is rotated about the a) x-axis b) the line y = 6 c) the line y = 8 d) the y-axis e) the line x = 3 f) the line x = 4 Example ) Find the volume if the region enclosing y = x, y = 0, x = is rotated about the a) x-axis b) the line y = 4 c) the line y = 5 d) the y-axis e) the line x = f) the line x = 4 MasterMathMentor.com Stu Schwartz

2 Example 3) Find the volume if the region enclosing y = 1" x, y = 1, x = 4 is rotated about the a) x-axis b) the line y = 3 c) the line y = 5 d) the y-axis e) the line x = 4 f) the line x = 6 Example 4) Find the volume if the region enclosing y = x, y = 3 x is rotated about the a) x-axis b) the y-axis c) the line y = 1 MasterMathMentor.com Stu Schwartz

3 Example 5) Find the volume of the solid whose base is bounded by the circle x cross sections taken perpendicular to the x-axis " y = 5 with the indicated a) squares b) equilateral triangles c) semi-circles d) isosceles right triangles Example 6) Find the volume of the solid whose base is bounded by the lines y = x! 4, y = 4! x and x = 0 with the indicated cross sections taken perpendicular to the x-axis a) squares b) equilateral triangles c) semi circles d) isosceles right triangles Example 7). A inch by inch by 4 inch wooden block is carved into the shape shown on the right below. The graph of y = 4! x is drawn on the back of the block. The wood is shaved off the front and top faces in such a way that the remaining solid has square cross sections perpendicular to the x-axis. Find the volume of the solid. y x 4 MasterMathMentor.com Stu Schwartz

4 Example 8). A horn for a public address system is to be made with the inside cross sections increasing exponentially with distance from the speaker. The horn will have the shape of the solid formed when the region bounded by y = e 0. 4 x and y = x "1 from x = 0 to x = 3 is rotated about the x-axis. If x and y are in feet, find the volume of the material used to make this speaker. Example 9) Prove that if you rotate the parabola y = ax around the y-axis, the volume of the resulting paraboloid will always be one-half of the circumscribed cylinder. MasterMathMentor.com Stu Schwartz

5 Volume by Disks/Washers - Homework 1. Find the volume if the region enclosing y = 4! x, x = 0, y = 0 is rotated about the a) x-axis b) the line y = 4 c) the line y = 5 d) the y-axis e) the line x = 4 f) the line x = 6. Find the volume if the region enclosing y = x " x, y = 0, x = is rotated about the a) x-axis b) the line y = 6 c) the line y = 9 MasterMathMentor.com Stu Schwartz

6 3. Find the volume if the region enclosing y = 1" x, x = 0, y = 0, x = 9 is rotated about the a) x-axis b) the line y = 4 c) the line y = 5 d) the y-axis e) the line x = 9 R = r = R = r = V = V = 4. Find the volume if the first quadrant region y = sin x and y = cos x on 0," 4 a) x-axis b) the y-axis # $ is rotated about 5. Find the volume of the solid whose base is bounded by the circle whose center is the origin and whose radius is 10 with the indicated cross sections taken perpendicular to the x-axis a) squares b) equilateral triangles c) semi circles d) isosceles right triangles MasterMathMentor.com Stu Schwartz

7 6. Find the volume of the solid whose base is bounded by the curves y = x! x! 3 and y = x with the indicated cross sections taken perpendicular to the x-axis a) squares b) equilateral triangles c) semi circles d) isosceles right triangles 7. A pyramid has a square base 8 cm by 8 cm and 8. The figure below shows a horn-shaped solid altitude 15 cm. Each section perpendicular to formed in such a way that a plane perpendicular the y-axis is a square. Find the volume of the to the x-axis cuts a circular cross-section. Each pyramid. Show that it equals one-third the circle has its center on the graph of y = 0. x and volume of the circumscribed rectangular a radius on the graph of y = 0. 16x + 1. Find the solid with the same base and altitude. volume of the solid if x and y are in centimeters. 15 cm y y # x, y$ 8 cm x x 8 cm BC Solutions Illegal to post on Internet

8 1 9. Prove that the volume of a right circular cone of base radius r and altitude h is given by V = " r h. To do this, 3 draw the cone on the graph below placing the center of the base of the cone at the origin. You may either draw the cone parallel to the x-axis or y-axis. 10. The countries of Appree and Hensive are spying on each other. The Apprees find that the Hensives are designing a new submarine. The cross section perpendicular to the horizontal axis of the sub will be circles with centers on that axis. The only quantitative piece of information that the Hensives have found is that the radius y at any distance x from the bow (front) of the sub is given by: y = 1 x! 0 0x where x and y are in meters. Your job is to find out as much about the sub as possible from this information. a) On the grid, sketch the sub. b. The sub will end where the radius becomes negative. How long is the submarine? Compare it to the length of a football field. c. What is the beam (maximum diameter) of the sub? How far from the bow is this maximum diameter? d. Fast submarines have a length-to-beam ratio of 7 or more. Is the submarine a fast sub? e. What is the total volume of the submarine? Show your integral. f. The displacement of a ship is the number of tons the ship weighs. The term displacement is used because a floating ship weight T tons will displace T tons of water. Given that a cubic meter is about 104 kg, how many metric tons will the Green September displace? (A metric ton is about 1000 kg). MasterMathMentor.com Stu Schwartz