7 3 VOLUMES WITH KNOWN CROSS SECTIONS VOLUMES

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7. 3 VOLUMES WITH KNOWN CROSS SECTIONS

7. 3 VOLUMES WITH KNOWN CROSS SECTIONS

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x 2 + y 2 = 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

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

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x 2 + y 2 = 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 3 -3 dx y-coordinate x So, s = 2 y

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x 2 + y 2 = 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 Volume of solid: 3 -3 dx x

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x

VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x 2 + y 2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. Volume of solid: 3 -3 y 3 -3 dx x

KNOWN CROSS SECTIONS Ex: The base of a solid is the region enclosed by

KNOWN CROSS SECTIONS Ex: The base of a solid is the region enclosed by the ellipse 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

5 1. ) Find the area of the cross section A(x). -2 a a y 2 -5 2. ) Set up & evaluate the integral.

EXAMPLE The base of a solid is the region enclosed by the triangle whose

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. Area of cross section (semicircle)? y r is half of the yvalue on the line 2 4 x

EXAMPLE The base of a solid is the region enclosed by the triangle whose

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. Area of cross section (semicircle)? y 2 Volume 4 x (f. Int) V = 2. 094

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