7.2 Volumes Using Cross Sections Name: 7.4 Volumes Using Shells Date: AP Calculus BC Period:
For questions #1-7, draw a picture, draw a sample cross section and label your sample points.
1. The base of a solid is a triangular region bounded by the coordinate axes and the line y3 x . Each slice of the solid perpendicular to the x-axis is a square. Find the volume of the solid.
2. Find the volume of the solid whose base is bounded by the circle x2 y 2 9 , and has cross sections perpendicular to the x-axis that are equilateral triangles.
3. The next base is bounded in the first quadrant by the y-axis, the parabola y x 2 , and the line 2x y 3 0 . Find the volume of the solid whose cross sections perpendicular to the x-axis are semi-circles.
4. A pyramid has a square base 8-cm by 8-cm and altitude 15-cm. Each cross section perpendicular to the y-axis is a square. 1. On a sketch of the pyramid, draw a representative cross section formed by slicing the solid perpendicular to the y-axis. (Hint: Place peak of pyramid at (0,15)) 2. Label 2 sample points. One is (x, y) in the cross section on the line segment connecting the vertex of the pyramid to the positive x-axis. Hint: you need an equation that guides the sample point (x, y). 3. Find the volume of the solid using the Fundamental Theorem of Calculus 4. Finally, show that finding the volume using the geometry volume formula for pyramids calculates the same answer as using calculus.
5. Region R under the graph of y= x 0.6 from x = 0 to x = 4 forms the back side of a solid. Cross sections of the solid perpendicular to the x-axis are isosceles right triangles with their right angle on the x-axis. a. Find the volume of the solid algebraically using the fundamental theorem. b. Next, state what the volume of the solid would be if the cross sections were squares instead of right triangles.
1 6. Rotate about the x-axis, the region in Quadrant I above the graph of y = that is bounded by x 2 the lines x = 5 and y = 4 . Use Shells.
7. Rotate about the y-axis the region bounded by the graph of y= x 2 -6x + 7 and the line x- y = -1 .
You can check your final answers on my website. The computer has a second page with answers.
Kruzich 14 Answers
1. 9
2. 36 3
53 3. 120
4. c. 320
5 5. a. 42.2 5b. 9.5964 22
6. 217.8254
875 7. p 458.1489 6
Kruzich 14