volume_part2vero - Volumesbyslicing*sec0on6.2...

Info icon This preview shows pages 1–8. Sign up to view the full content.

Volumes by slicing *sec0on 6.2*
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Choose an axis (e.g., the x‐axis) Cut your solid in slices perpendicular to this axis Δx= the thickness of a slice A(x)= the cross‐sec0onal area of a slice Add the volume of all the slices Riemann sum: Σ ΔV = Σ A(x) Δx Send the number of slices to infinity the Integral of A(x) in dx. A(x) ΔV = A(x) Δx thickness of slice cross‐sec/onal area of slice volume of slice The volume of a slice is ΔV = A(x) Δx “the philosophy of slicing”
Image of page 2
The volume of a cylinder Cylinder with radius r and height h Every cross‐sec0on is a circle of radius r. A(x)=r 2 V = A ( x ) dx = π r 2 dx = π r 2 1 dx = π r 2 h 0 h 0 h 0 h
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Pyramid with square basis Side=L Height =h A(x)=s 2 The volume of a pyramid (s/2) : (L/2) = x : h L’ P’ s = (L/h) x A(x)=(L 2 /h 2 )x 2 The triangles OPL and OP’L’ are similar We need to find s (as a func0on of x). Every cross‐sec0on is a square. V = A ( x ) dx = L 2 h 2 0 h 0 h x 2 dx = L 2 h 2 x 2 0 h dx = L 2 h 2 h 3 3 = L 2 h 3 L
Image of page 4
The volume of a frastum of a pyramid d V = A ( x ) dx 0 h A(x) V = A ( x ) dx 0 d V = A ( x ) dx h d h A) B)
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

The volume of a frastum of a pyramid d V = A ( x ) dx 0 h A(x) V = A ( x ) dx 0 d V = A ( x ) dx h d h A) B) OK
Image of page 6
The volume of a cone d V = A ( x ) dx = 3 π 0 h TRUE FALSE A) B) A(x)
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 8
This is the end of the preview. Sign up to access the rest of the document.
  • Fall '09
  • FAMIGLIETTI,C
  • Calculus, dx, v=

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern