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Calculus Section 7.2 Volume by Cross Sections
-Find the volume of a solid with known cross sections
Homework: page 465 #’s
61ab, 62c, 63a
With the disk and washer method, we’ve found volumes of areas having circular cross sections with area
A =πR2. This method can be generalized to solids of any shape as long as you can choose a cross section with a
known formula for the area. Some common cross sections are squares, rectangles, triangles, and trapezoids.
Volumes of Solids with Known Cross Sections
1) For cross sections of area A(x) taken perpendicular to the x-axis,
Volume =
 A( x)dx
2) For cross sections of area A(y) taken perpendicular to the y-axis,
Volume =
 A( y)dy
Example) Triangular Cross Sections
Find the volume of a triangular shaped solid whose base is the region bounded by the lines f(x) = 1 – x/2,
g(x) = -1 + x/2, and x = 0. The cross sections perpendicular to the x-axis are equailateral triangles.
Example) Square Cross Sections
Find the volume of the solid whose base is bounded by the circle x2 + y2 = 4 with square cross sections taken
perpendicular to the y-axis.