 The volume of a known integrable cross- section area A(x) from x = a to x = b is  Common areas:  square: A = s 2 semi-circle: A = ½  r 2 equilateral.

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Presentation transcript:

 The volume of a known integrable cross- section area A(x) from x = a to x = b is  Common areas:  square: A = s 2 semi-circle: A = ½  r 2 equilateral triangle: A =  3/4 s 2

 The base of a solid is the region enclosed by y = 9 – x 2 and the x- axis. Cross sections, taken perpendicular to the x-axis are squares. Find the volume of the solid.

 Same base.  Cross sections are semi-circles. Find the volume of the solid.

 The base of a solid is the region enclosed by the graphs of y = x 2, y = 9 – x 2 and x = 0 in the 1 st quadrant. Cross sections perpendicular to the x-axis are squares. Find the volume.

 Same base.  Cross sections are equilateral triangles. Find the volume of the solid.

 The base of a solid is the region enclosed by the graphs of y = x, y = – x and x = 3. Cross sections taken perpendicular to the x- axis are equilateral triangles. Find the volume of the solid.

 Same base.  Cross sections are semi-circles. Find the volume of the solid.

 The base of a solid is the region enclosed by y = 1/x, y = 0, x = 1 and x = 4. Cross-sections taken perpendicular to the x-axis are isosceles right triangles with the hypotenuse across the base. Find the volume.

1/x

 Let R be the region in the first quadrant enclosed by the graphs of y=2x and y=x 2, as shown in the figure.

a) Find the area of R. b) The region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x-axis has area. Find the volume of the solid. c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.