Volumes of Solids with Known Cross Sections

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Presentation transcript:

Volumes of Solids with Known Cross Sections 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

Write 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

Write 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

Steps 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 direction If shapes are vertical, you will use dx If shapes are horizontal, you will use dy Re-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 shape Integral: Solve the Integral (use your calculator unless it is a very simple integral)

The base of a solid is bounded by. and x = 9 The base of a solid is bounded by and x = 9. The cross sections of the solid taken perpendicular to the x-axis are squares.

The 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.

The 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.

A 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.

The base of a solid is the region in the first quadrant bounded by the graph of y = 3x1/2 – x3/2 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?

The base of a solid is the elliptical region with boundary curve 9x2 + 4y2 = 36. Cross sections perpendicular to the x-axis are rectangles with height 3. Find the volume of the solid.

The base of a solid is a region bounded by the curves y = x2 and y = 1 The base of a solid is a region bounded by the curves y = x2 and y = 1. Cross sections perpendicular to the y-axis are semicircles. Find the volume of the solid.

The base of a solid is the region bounded by the graphs of x2 = 16y and y = 2. Cross sections perpendicular to the x-axis are rectangles whose heights are half the size of the base.