Area between curves and volumes of solids

Learning objectives

Set up and evaluate the area between two curves as the integral of the upper minus the lower function with respect to x, or the right minus the left function with respect to y

Split the integral at the intersection points when the curves change relative position, and find those intersections with technology in calculator-active questions

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

Apply the disc method with V = π∫[r(x)]² dx to a solid of revolution, adjusting the radius when the axis of revolution is a line other than a coordinate axis

Apply the washer method with V = π∫([R]² − [r]²) dx when the region does not touch the axis of revolution, identifying the outer and inner radii correctly

Choose whether to integrate with respect to x or y so that the region is described by a single integral, and set up the integral from a sketch even when it will not be evaluated by hand