Ms Battaglia AP Calculus Write an equation to give the area of each shape Square Equilateral Triangle Rectangle that is three times Isosceles Triangle that is twice as tall as it is wide as high as its base ID: 919301
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Slide1
Volumes of Solids with Known Cross Sections
Ms. Battaglia
AP Calculus
Slide2Write an equation to give the area of each shape.
Square Equilateral Triangle
Rectangle that is three times Isosceles Triangle that is twice
as tall as it is wide as high as its base
Slide3Write an equation to give the area of each shape.
Semicircle Isosceles Triangle
Rectangle that is twice Rectangle that is two-thirds
as wide as it is tall as tall as it is wide
Slide4Write an equation to give the area of each shape in terms of x.
Quarter Circle Quarter Circle
Isosceles triangle that is
Equilateral Triangle twice as tall as it is wide
Slide5Steps for Finding Volume of a Solid with Known Cross Sections
Draw the graph(s) that define the base
Determine any points of intersection and/or zeros
Sketch a representative cross section: pay attention to directionIf shapes are vertical, you will use dxIf shapes are horizontal, you will use dyRe-write equations and find your bounds in terms of x or y (match the dx or
dy)Write an equation for the length of the line that determines the bottom of each shape
Write an equation for the area of the shapeIntegral: Solve the Integral (use your calculator unless it is a very simple integral)
Slide6The base of a solid is bounded by and x = 9. The cross sections of the solid taken perpendicular to the x-axis are squares.
Slide7The base of a solid is enclosed by y = x, y = 5, and x = 0 and the cross sections are semicircles, perpendicular to the x-axis. Find the volume of the solid.
Slide8The base of a solid is the region enclosed by a circle centered at the origin with a radius of 5 inches. Find the volume of the solid if all cross sections perpendicular to the x-axis are squares.
Slide9A mathematician has a paperweight made so that its base is the shape of the region between the x-axis and one arch of the curve y = 2sinx. Each cross-section perpendicular to the x-axis is a semicircle whose diameter runs from the x-axis to the curve. Find the volume of the paperweight.
Slide10The base of a solid is the region in the first quadrant bounded by the graph of y = 3x
1/2
– x
3/2 and the x-axis. Cross sections perpendicular to the x-axis are isosceles right triangles, with one leg on the xy-plane. What is the volume of the solid?
Slide11The base of a solid is the elliptical region with boundary curve
9x
2
+ 4y2 = 36. Cross sections perpendicular to the x-axis are rectangles with height 3. Find the volume of the solid.
Slide12The base of a solid is a region bounded by the curves y = x
2
and y = 1. Cross sections perpendicular to the y-axis are semicircles. Find the volume of the solid.
Slide13The base of a solid is the region bounded by the graphs of
x
2
= 16y and y = 2. Cross sections perpendicular to the x-axis are rectangles whose heights are half the size of the base.