Volumes by cross-sections, discs and washers

Learning objectives

Find the volume of a solid whose base is a plane region and whose cross-sections perpendicular to an axis are squares, rectangles, triangles or semicircles, by integrating the cross-sectional area

Use the disc method V = π ∫ [r(x)]² dx to find the volume of a solid of revolution about the x-axis or the y-axis

Use the disc method to find the volume of a solid formed by revolving a region about a horizontal or vertical line other than an axis, adjusting the radius accordingly

Use the washer method V = π ∫ ([R]² − [r]²) dx to find the volume of a solid of revolution with a hole, about an axis or about another line

Set up, and where required evaluate, volume integrals from a sketch of the region, choosing the variable of integration to match the axis of revolution or the direction of the cross-sections