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1. DEFINION OF THE VOLUME OF A SOLID Consider a solid S that lies between a = a and a: = b. Let A(x) denote the crosssectional area of S in the plane through a: and perpendicular to the :c-axis. If A is a continuous function, then the volume of S is n b V: 5130 ;A($,;)Azt = f A(a:)da;. 2. SOLIDS OF REVOLUTION Theorem 1 (Volume of the solid of revolution - volume by cross-section). Let f (3:) 2 9(35) 2 0 on the interval [a, b]. Let R be the region about the y = f (it) and y = g(:1:) between a: = a and a: = b. The volume of the of the solid obtained by rotating about the m-axis of the region R is v = r ] [f(:v)2 gedx. 3. HOMEWORK (1) Find the volume of the solid obtained by rotating the region bounded by the curves y = 11:2 and y = and the lines 3: = 4 and :3 = 9 about the :caxis. (2) Find the volume of the solid obtained by rotating the region bounded by y = x and y = about the m-axis. (3) Find the volume of the solid obtained by rotating the region bounded by y = a: and y = about the y-axis. (4) Find the volume of the solid obtained by rotating the region bounded by y = a: and y = about the y = 1. (5) Describe and sketch a picture of the solid whose volume is represented by the integral (a) 1r/09(81 932)dac. (b) 7T /1 3 ads- (6) Set up but do not evaluate the integral representing the volume of the region (circular ellipsoid) bounded by the ellipse as:2 + by2 = r2 about the x-axis