7.3 Volumes - Oxford School District

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. 3

-3 y 3 -3 dx Area of cross section (square)? 2 A s 2 A (2 y ) 2 A 4 y 2 x y 9 y 9 x y-coordinate So, s = 2y A 4 9 x x

2 2 A 4(9 x ) 2 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. 3 -3 Area of cross section 2 y (square)? A 4(9 x )

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 1 3 V 4 9 x x 3 3 3 -3 dx 3 x V 418 18 V 4 36 V 144 KNOWN CROSS SECTIONS Ex: The base of a solid is the region enclosed by x2 y2 the ellipse

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 2 2 a a (2 y ) 2 2 2a 4 y a 2 y

y 2 2 -5 1 2 A( x) a 2 1 A( x) 2 y 2 A( x) y 2 2.) Set up & evaluate the integral. 2 25 x 2 A( x) 25 4 2 2 200

25 x 3 units 25 dx 3 4 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 mx 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 Volume 4 2 2 1 1 V x 1 dx 2 0 4 4 x (fInt) V = 2.094