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6 049 Volume

6 049 Volume. AP Calculus. Volume. Volume = the sum of the quantities in each layer. where h is # layers. x. x. x. x. y. x-axis. Volume by Cross Sections. n. foundation. BE7250.

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6 049 Volume

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  1. 6049 Volume AP Calculus

  2. Volume Volume = the sum of the quantities in each layer where h is # layers

  3. x x x x y x-axis

  4. Volume by Cross Sections n foundation

  5. BE7250 Axial (cross sectional) magnetic resonance image of a brain with a large region of acute infarction, formation of dying or dead tissue, with bleeding. This infarct involves the middle and posterior cerebral artery territories. Credit: Neil Borden / Photo Researchers, Inc.

  6. (Finding the volume of a solid built on the base in the x – y plane) METHOD: 1.) Graph the “BASE”  2.) Sketch the line segment across the base. That is the representative slice “n” Use “n” to find: a.) x or  y (Perpendicular to axis) b.) the length of “n”  3.) Sketch the “Cross Sectional Region” - the shape of the slice (in 3-D ) from Geometry V = B*h h = or is the thickness of the slice B = the Area of the cross section  4.) Find the area of the region  5) Write a Riemann’s Sum for the total Volume of all the Regions Volume by Slicing

  7. The base of a solid is the region in the x-y plane bounded by the graph andthe y – axis. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a square. Example 1: Base h Cross -Section

  8. The base of a solid is the region in the x-y plane bounded by the graph andthe y – axis. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base). Example 2: h n n

  9. Some Important Area Formulas Square- side on base Square- diagonal on base n n Equilateral Δ Isosceles rtΔ leg on base Isosceles rtΔ hypotenuse on base n n n Semi - Circle Circle diameter on the base n n

  10. The solid lies between planes perpendicular to the x - axis at x = -1 and x = 1 . The cross sections perpendicular to the x – axis are circular disks whose diameters run from the parabola y = x2 to the parabola y = 2 – x2. EXAMPLE #4/406 n

  11. The base of a solid is the region bounded by The cross sections, perpendicular to the x-axis, are rectangles whose height is 3x

  12. Assignment: P. 406 # 1 - 6 all If the problem has multiple parts work “ a “ then set up only ( to the definite integral) the other parts.

  13. Volumes of Revolution:Disk and Washer Method AP Calculus

  14. Volume of Revolution: Method Lengths of Segments: In revolving solids about a line, the lengths of several segments are needed for the radii of disks, washers, and for the heights of cylinders. A). DISKS AND WASHERS 1) Shade the region in the first quadrant (to be rotated) 2) Indicate the line the region is to be revolved about. 3) Sketch the solid when the region is rotated about the indicated line. 4) Draw the representative radii, its disk or washer and give their lengths. <<REM: Length must be positive! Top – Bottom or Right – Left >> Ro = outer radius ri = inner radius

  15. Disk Method Rotate the region bounded by f(x) = 4 – x2in the first quadrant about the y - axis adjacent to The region is _______________ _______ the axis of rotation. The Formula: The formula is based on the _____________________________________________ Volume of a cylinder Right-left

  16. Washer Method Rotate the region bounded by f(x) = x2, x = 2 , and y = 0 about the y - axis Separate from The region is _______________ __________ the axis of rotation. The Formula: The formula is based on _____________________________________________ Big cylinder-small cylinder

  17. Disk Method Rotate the region bounded by f(x) = 2x – 2 , x = 4 , and y = 0 about the line x = 4 Adjacent to The region is _______________ _______ the axis of rotation.

  18. Washer Method Rotate the region bounded by f(x) = -2x + 10 , x = 2 , and y = 0 about the y - axis Separate from The region is _______________ __________ the axis of rotation. Separate V=

  19. The region is bounded by Rotated about: the x-axis, and the y-axisa) The x-axis b) The y-axis c) x = 3 d) y = 4 Example 1:

  20. The region is bounded by: Rotated about: f(x) = x and g(x) = x2 a) the x-axis in the first quadrant b) the y-axis c) x = 2 d) y = 2 Example 2:

  21. The base of a solid is the region in the x-y plane bounded by the graph andthe x – axis. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base). Example 2:

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