SECTION 7-3-C Volumes of Known Cross - Sections

Recall: Perpendicular to x – axis Perpendicular to y – axis

Volumes with Known Cross-Sections No more revolving about an axis Solids are built from base formed by bounded regions Projected from this region are cross-sections of geometric shapes

Steps: To find Volume with a Known Cross- Section 1. Sketch the base and a typical cross-section 2. Find a formula for A(x) 3. Find the limits of integration 4. Integrate A(x) to find the volume

Cross Section formulas Squares Circles Semicircles Equilateral Triangles Isosceles Triangles

1) Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections are all squares perpendicular to the x – axis whose side lie on the circle

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2) Find the volume of a solid whose base is the region between x – axis and y = 4 – x 2 where the perpendicular cross sections are equilateral triangles with sides on the base

3) Find the volume of a solid whose base is the region between one arch of y = sin(x) and the x – axis where the cross sections perpendicular to the x - axis are equilateral triangles with sides on the base. 0 

4) Find the volume of the solid with a base in the first quadrant bounded by, with square cross sections perpendicular to the x – axis. 2 1

5) Find the volume of the solid with a base in the first quadrant bounded by whose cross sections are rectangles of height ¼ which are perpendicular to the y – axis. 2 1

Homework Worksheet 7-3-C Cross Sections