6.2C Volumes by Slicing with Known Cross-Sections

Known Cross Sections Method Volume can be calculated by finding area of known geometric shapes and multiplying by thickness (dx). Here is an example of squares stacked on top of a circular region.

Visualizations Rectangular Cross-Sections Semicircular Cross-Sections Equilateral Triangle Cross-Sections

We can find the area of each cross section, then add an infinite number of infinitely thin cross sections. When we multiply by thickness, we have volume.

Examples Cross sections may be rectangles, semi-circles or triangles. The base of the solid may be a rectangle, circle, triangle or an irregular shape. Mathematica

Method of Slicing: 1 Sketch the base of the solid (including a typical slice) and a typical cross section. Find a formula for A(x) and multiply by dx for width. 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with square cross sections perpendicular to the x-axis. y x 422/51(c)

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with semicircular cross sections perpendicular to the x-axis y x 422/51(c)

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with equilateral triangle cross sections perpendicular to the x-axis. x y

Find the volume of the solid formed with the region defined by and as the base and cross sections that are squares perpendicular to the base and the x-axis. 81/10 sq units

3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. h s This correlates with the formula: dh 3

p Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections p

Ex. Find the volume of the solid whose base is a circle of radius 1 in the first quadrant and with square cross-sections  x-axis.

Ex. Find the volume of the solid whose base is bounded by the x-axis, the y-axis, x = 9, and with semi-circular cross-sections  x-axis.

Find the volume of the solid whose base is a circle of radius 1 centered at the origin and with isosceles right triangles cross-sections  x-axis.