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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 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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7.3 Volumes With Known Cross Sections

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  1. 7.3 Volumes With Known Cross Sections

  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. 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

  4. 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

  5. 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

  6. 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

  7. 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

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

  9. 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

  10. 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

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