Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 331 APPLICATIONS OF INTEGRATION 11: VOLUME 259EXAMPLE Consider the hctiOn /(XI - ,&- for 0 s x 5 r. Thc graph of f is the part or the circleY' + y2 = r' that lies in the first quadrrnL Revolution about ihc y-axis of the region a under the graph orjlseeFb. 3%) prod\ica a solid heinisphm of radius r. By the cylindrical shell formula,To evaluate V substirutc U = rz - xz. Then du = -2x dx, and the limits or integration x = 0 and x = rkoome U = rz and U = 0. respectively,vhir result is more easily obtained by the disk rormula V = A3.2 VOLUME BASED ON CROSS SECTIONSUrx2 dy. Try it.)-Assume that a salid (not nacxssarily a solid of wolution) lia entirely between the plane perpendicularto the x-axiu at x and the plane perpendicular to the x-axis at x = b. For a 5 x 5 b. let thepllne perpendicular to the x-axis at that value or x intersect the solid in a region of area A(x), asindicated in Fig. 33-7. Then thc volume V of the solid is given bydxcross-sectionformdoFor il derivation, see Problem 33.6.