Volume of Regions with cross- sections ----- an off shoot of Disk MethodV =  b a (π r 2 ) dr Area of each cross section (circle) * If you know the cross.

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Volume of Regions with cross- sections an off shoot of Disk MethodV =  b a (π r 2 ) dr Area of each cross section (circle) * If you know the cross section’s shape, then V= A(x) dx where A(x) is the cross section’s area.  b a

Steps for success with Known Cross Sections 1. Draw the graph of the base on the coordinate plane and darken a rectangle (slash) on it. 2a. Determine the length of that rectangle ( top-bottom or right-left ) 2b. Draw a cross-section and put length from #2a on it ( note its location from the original description). Find an equation for the area of this cross- section using the length on the figure (Final formula should be in terms of one variable). 3. Integrate the area formula using boundaries from base drawing (#1).

Let R be the shaded region bounded by the graphs of y = x and y = e - 3x and the vertical line x = 1 as shown in the figure above ( c ) The region R is the base of a solid. For this solid, each cross section perpendicular to the x- axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. 1 1 y x O

The base of a certain solid is the circle x 2 + y 2 = 9 and each cross- section perpendicular to the x-axis is an equilateral triangle with one side across the base. Find the volume. Problem set on back page # 2a