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7.3 VOLUMES WITH KNOWN CROSS SECTIONS VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. SOLIDS WITH KNOWN CROSS SECTIONS If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the solid from x = a to x = b is b V A( x)dx a VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. Area of cross section (square)? 3 -3 3 -3 dx As 2 A (2 y ) 2 A 4y y 2 y-coordinate So, s = 2y A 4 9 x x 2 A 4(9 x ) 2 x y 9 2 2 y 9 x 2 2 VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. Area of cross section (square)? 3 -3 y A 4(9 x ) 2 Volume of solid: x2 V A( x)dx x1 3 -3 dx x 3 V 4(9 x 2 ) dx 3 VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. Volume of solid: 3 3 -3 V 4(9 x 2 ) dx y 3 3 V 4 (9 x 2 ) dx 3 3 1 3 V 49 x x 3 3 3 -3 dx x V 418 18 V 436 V 144 KNOWN CROSS SECTIONS Ex: The base of a solid is the region enclosed by the 2 y2 ellipse x 1 4 25 The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid. 5 -2 a a 2 -5 5 1.) Find the area of the cross section A(x). -2 a a a a (2 y ) 2 2 2a 4 y a 2 y 2 y 2 2 25x 2 A( x ) 25 4 2 -5 1 2 A( x ) a 2 1 A( x ) 2 y 2 A( x ) y 2 2 2.) Set up & evaluate the integral. 25x 2 200 units 3 2 25 4 dx 3 2 EXAMPLE The base of a solid is the region enclosed by the triangle whose vertices are (0, 0), (4, 0), and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Find the volume of the solid. y Area of cross section (semicircle)? 1 2 A r 2 2 4 1 1 1 A x 2 2 2 2 x r is half of the yvalue on the line y m x b 2 1 1 A x 1 2 4 y 2 1 x2 2 EXAMPLE The base of a solid is the region enclosed by the triangle whose vertices are (0, 0), (4, 0), and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Find the volume of the solid. y Area of cross section (semicircle)? 1 1 A x 1 2 4 2 2 Volume 2 1 1 V x 1 dx 2 0 4 4 4 x (fInt) V = 2.094