1 / 12

The region enclosed by the x -axis and the parabola

DO NOW. The region enclosed by the x -axis and the parabola. is revolved about the line x = –1 to. generate the shape of a cake. What is the volume of the cake?. The Shell Method. Section 7.3d. The region enclosed by the x -axis and the parabola. is revolved about the line x = –1 to.

kitra
Download Presentation

The region enclosed by the x -axis and the parabola

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. DO NOW The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake?

  2. The Shell Method Section 7.3d

  3. The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? Let’s first try to integrate with respect to y Washers!!! x = –1 To find the inner and outer radii of our washer, we wouldneed to get the equation of the original parabola in terms of x... not an easy task…

  4. The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? Instead of slicing horizontally, we will cut a series of cylindrical slices by cutting straight down all the way around the solid. The radii of the cylinders gradually increase, and the heights of the cylinders follow the contour of the parabola: smaller to larger, then back to smaller. When we unroll one of these shells, we have a rectangular prism, with height equal to the height of the shell, length equal to the circumference of the shell, and thickness equal to .

  5. The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? Volume of each shell: Here, we have To find the total volume of the solid, integrate with respect to x:

  6. The region bounded by the curve , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid. Let’s solve this problem first by integrating with respect to x: Cross section area: Volume:

  7. The region bounded by the curve , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid. Now let’s integrate with respect to y using our new technique: Radius of each shell: 0 to 2 Limits of integration: Height of each shell: Thickness of each shell:

  8. The region bounded by the curve , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid. Now let’s integrate with respect to y using our new technique: Volume:

  9. Let’s redo a problem from a previous class, this time using shells… Find the volume of the solid generated by revolving the region bounded above by the square root function and below by the identity function about the y-axis Our previous solution Cross section area: Volume:

  10. Let’s redo a problem from a previous class, this time using shells… Find the volume of the solid generated by revolving the region bounded above by the square root function and below by the identity function about the y-axis Shell radius: Shell height: Shell thickness: Volume:

  11. Let’s redo a problem from a previous class, this time using shells… Find the volume of the solid generated by revolving the triangular region bounded by the lines y= 2x, y = 0, and x = 1 about the line x = 2. Our previous solution Cross section area: r R Volume:

  12. Let’s redo a problem from a previous class, this time using shells… Find the volume of the solid generated by revolving the triangular region bounded by the lines y= 2x, y = 0, and x = 1 about the line x = 2. Shell radius: Shell height: Shell thickness: Volume:

More Related