Section 7. Volume: The Disk Method
White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3
White Board Challenge Calculate the volume V of the solid obtained by rotating the region between y = 5 and the x-axis about the x-axis for 1 x 7. No Calculator V radius height 5 V 5 6 6 V 150 V 471.39
Volumes of Solids of Revolution with Riemann Sums The Riemann Sum is set up by considering this cross sections of the solid (circles) each with thickness dx: a x k Radius b Volume n lim max x k 0 k 1 b a radius x k radius dx b a radius dx
Volumes of Solids of Revolution: Disk Method Sketch the bounded region and the line of revolution. (Make sure an edge of the region is on the line of revolution.) If the line of revolution is horizontal, the equations must be in y= form. If vertical, the equations must be in x= form. Sketch a generic disk (a typical cross section). Find the length of the radius and height of the generic disk. Disk Method = Integrate with the following formula: No hole in the b V radius height a solid.
Line of Rotation Example 1 Calculate the volume of the solid obtained by rotating the region bounded by y = x and y=0 about the x-axis for 0 x. Sketch a Graph Find the Boundaries/Intersections Radius = x x 0, Integrate the Volume of Each Generic Disk 0 x dx Height = dx Make Generic Disk(s) 3 5
Line of Rotation Example Calculate the volume of the solid obtained by rotating the region bounded by y = x and y=4 about the line y = 4. Sketch a Graph Find the Boundaries/Intersections x 4 4 - x x, Integrate the Volume of Each Radius = Height = dx Make Generic Disk(s) Generic Disk 4 x dx 51 15 NOTE: Because of the square, the order of subtraction does not matter.
Example 3 Calculate the volume of the solid obtained by rotating the region bounded by y = x, x=0, and y=4 about the y-axis. Make Generic Disk(s) Sketch a Graph Radius = y Height = dy Line of Rotation Find the Boundaries/Intersections Remember: 0 x Integrate the Volume of Each Generic Disk Since the Line of Revolution is Vertical, Solve for x 0 y x x x 4 y y dy We only need 0 x y 0 y 4 y 0 8
White Board Challenge Find the volume of the following three-dimensional shape: 6 ft ft No Calculator 1 ft V 3 1 1 1 3 1 1 96 301.593 ft 3
White Board Challenge Calculate the volume V of the solid obtained by rotating the region between y = 5 and the y = about the x-axis for 1 x 7. V r h r h outer inner No Calculator 5 6 V V outer inner V r r h 5 16 V 395.841 6
Area of a Washer The region between two concentric circles is called an annulus, or more informally, a washer: outer inner Area R R R inner outer inner Area R R R outer
Volumes of Solids of Revolution: Washer Method Sketch the bounded region and the line of revolution. If the line of revolution is horizontal, the equations must be in y= form. If vertical, the equations must be in x= form. Sketch a generic washer (a typical cross section). Find the length of the outer radius (furthest curve from the rotation line), the length of the inner radius (closest curve to the rotation line), and height of the generic washer. Integrate with the following formula: b a outer inner V r r height Always a difference of squares. Washer Method = Hole in the solid.
Line of Rotation Example 1 Calculate the volume V of the solid obtained by rotating the region bounded by y = x and y=0 about the line y = - for 0 x. Sketch a Graph Find the Boundaries/Intersections R inner = 0 - - = R outer = x - - = x + Height = dx Make Generic Washer(s) x 0, Integrate the Volume of Each Generic Washer x 0 56 15 dx
Line of Rotation Example Calculate the volume V of the solid obtained by rotating the region bounded by y = e x and y= (x +) about the line y =. Sketch a Graph R outer = - e x R inner = - (x +) Find the Boundaries/Intersections x e x x 1.981, 0.448 Integrate the Volume of Each Generic Washer 0.448 x 1.981 e x dx Height = dx 8.536 Make Generic Washer(s)
Warm-up : 1985 Section I CAN DO NOW: No Calculator
Volume of a Right Solid A right solid is a geometric solid whose sides are perpendicular to the base. The volume of a right solid is the area of the base times the height. H Solid Volume A H Base Solid A Base
Volumes of Solids: Slicing Method Sketch the bounded region. If the cross section is perpendicular to the x-axis, the equations must be in y= form. If the y-axis, the equations must be in x= form. Sketch a generic slice (a typical cross section). Find the area of the base and the height of the generic slice. Integrate with the following formula: Must Answer #: How does the length across the bounded region help find the area of the base of the generic slice? b V A height a Base Must Answer #1: What does the length across the bounded region represent in your generic slice?
Side Length Example 1 Find the volume of the solid created on a region who base is bounded by y = x and the x-axis for 0 x 9. Let each cross section be perpendicular to the x-axis and be a square. Sketch a Graph Find the Boundaries/Intersections x 0,9 Height = dx A Base = A Square = side = ( x) Integrate the Volume of Each Generic Slice 9 0 x dx Make Generic Slice(s) 81
Diagonal Example Find the volume of the solid created on a region who base is bounded by x + y = 1. Let each cross section be perpendicular to the x-axis and be a squares with diagonals in the xy-plane. Height = dx If diameter is known, a side length is d d Sketch a Graph Make Generic Slice(s) A Base = A Square = side Since the Cross Sections are Per. to the x-axis, solve for y 1 x 1 x 1x Find the Boundaries/Intersections x 1,1 Integrate the Volume of Each Generic Slice 1 1 x y 1 y 1 1 x x dx 8 3
White Board Challenge A solid has base given by the triangle with vertices (-4,0), (0,8), and (4,0). Cross sections perpendicular to the y-axis are semi-circles with diameter in the plane. Calculator What is the volume of the solid? x 1 y4 A Base = ½πr x Diameter 1 y 4 Height = dy Radius = -½y+4 8 1 1 4 0 y 64 3 dy