1 / 14

Areas and Volumes

Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College. Photo by Vickie Kelly, 2003. Areas and Volumes. Gateway Arch, St. Louis, Missouri. We could split the area into several sections, use subtraction and figure it out, but there is an easier way.

jlisk
Télécharger la présentation

Areas and Volumes

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. Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, 2003 Areas and Volumes Gateway Arch, St. Louis, Missouri

  2. We could split the area into several sections, use subtraction and figure it out, but there is an easier way. How can we find the area between these two curves?

  3. Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx.

  4. Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips, we get:

  5. The formula for the area between curves is: We will use this so much, that you won’t need to “memorize” the formula!

  6. If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

  7. We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip

  8. 1 General Strategy for Area Between Curves: Sketch the curves. Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area. p

  9. 3 3 3 0 h 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. This correlates with the formula: s dh

  10. 1 Method of Slicing: Sketch the solid and a typical cross section. Find a formula for V(x). (Note that I used V(x) instead of A(x).) 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.

  11. y x If we let h equal the height of the slice then the volume of the slice is: h 45o x A 45o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. Since the wedge is cut at a 45o angle: Since

  12. y x Even though we started with a cylinder, p does not enter the calculation!

  13. Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections p

More Related