Solids not generated by Revolution Volumes Using Cross-Sections Solids Solids of Revolution Solids not generated by Revolution

Volumes Using Cross-Sections The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid Example: 2 Cross-sections: Base: is bounded by the curve and the line y =2 If the cross-sections of the solid perpendicular to the y-axis are squares

VOLUMES the cross-sections of the solid perpendicular to the y-axis are squares The base of a solid is bounded by the curve and the line y =2 https://www.youtube.com/watch?v=r545xs2i_pc

VOLUMES Example: Base: Cross-sections: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Example: Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

VOLUMES Example: Base: Cross-sections: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Example: Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

VOLUMES Example: Base: Cross-sections: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Example: Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

VOLUMES Example: Base: Cross-sections: The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi. If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles. Example: Base: is bounded by the curve and the line y =0 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles

Volumes Using Cross-Sections step1 Graph and Identify the region ( graph with an angle) The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid Example: 2 step2 Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem) step3 Find the length (S)of the segment from the two intersection points with the boundary Cross-sections: If the cross-sections of the solid perpendicular to the y-axis are squares step4 Cross-section type: Square  S = side length Semicircle  S = diameter Equilatral  S = side length step4 Cross-section type: Square  Semicircle  Equilatral  step6 Specify the values of x The volume is given by step7

VOLUMES Example: Cross-sections: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Example: step1 Graph and Identify the region ( graph with an angle) step2 Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem) Cross-sections: step3 Find the length (S)of the segment from the two intersection points with the boundary If the cross-sections of the solid perpendicular to the x-axis are semicircle step4 Cross-section type: Square  S = side length Semicircle  S = diameter Equilatral  S = side length step4 Cross-section type: Square  Semicircle  Equilatral  step6 Specify the values of x The volume is given by step7

VOLUMES T-102 Type equation here.Type equation here. here.       

VOLUMES T-132

VOLUMES T-122

VOLUMES T-092

VOLUMES 54.The base of a solid is a circular disk with radius r = 2. Parallel cross-sections perpendicular to the base are squares.

VOLUMES