Solid Volume Calculations with Different Cross-Sections
Learn how to find the volume of a solid with a base of x^2 + y^2 = 4 using various cross-sections: squares, equilateral triangles, semicircles, and isosceles right triangles. Understand the integration process and formulas involved.
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Presentation Transcript
Procedure: volume by slicing osketch the solid and a typical cross section o find a formula for the area, A(x), of the cross section o find limits of integration o integrate A(x) to get volume
Visualizations Rectangular Cross-Sections Semicircular Cross-Sections Equilateral Triangle Cross-Sections
Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle. First, find the length of a side of the square the distance from the curve to the x-axis is half the length of the side of the square solve for y + = = 2 2 x y 4 length of a side is : 2 2 2 y 4 x 2 4 x = 2 y 4 x ) ( ( ) 2 2 ( ) = 2 2 Area 2 4 x 4 4 x = 16 4 2 Volume x dx = 2 16 4 x 2
Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle. 4 ?2 x2 + y2 = 4 ? = ? = 2 4 ?2 ? = ?2 ?? = ? ?? 2 4 ?2?? =128 ? = 4 3 2
Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all equilateral triangles whose sides lie on the base of the circle. ?? = ? ?? ? = ? 4 ?2 x2 + y2 = 4 ? = 2 1 2? ? 2 3 4 ?2= 3 4 ?2 ?2 ? = = 2 3 4 ?2?? =32 ? = 3 18.475 2
Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all semicircles whose sides lie on the base of the circle. ? = ? 4 ?2 x2 + y2 = 4 ? = 8? ?2= ?4 ?2 2 1 2 ? ? 2 =1 ? = 2 ?? = ? ?? 2 ?4 ?2 ?? =16? ? = 16.755 2 3 2
Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are all Isosceles right triangles whose sides lie on the base of the circle. ? = ? 4 ?2 x2 + y2 = 4 ? = ? 2 = ?2 1 2? = 4 ?2 ? = ?4 tan 4 ?? = ? ?? 2 4 ?2 ?? =32 ? = 3 10.667 2
