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7.3 Volumes With Known Cross Sections. 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.
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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
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 Area of cross section (square)? -3 y 3 -3 y-coordinate So, s = 2y x dx
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 Area of cross section (square)? -3 y Volume of solid: 3 -3 x dx
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 y 3 -3 x dx
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
1.) Find the area of the cross section A(x). y 2.) Set up & evaluate the integral. 5 -2 a a 2 -5
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)? 2 r is half of the y-value on the line 4 x
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)? 2 Volume 4 x (fInt) V = 2.094