Volumes by Slicing

disk Find the Volume of revolution using the disk method washer Find the volume of revolution using the washer method shell Find the volume of revolution using the shell method known cross sections Find the volume of a solid with known cross sections Volume of REVOLUTION

These are a few of the many industrial uses for volumes of revolutions.

Volume of the washer R r

Examp le: Find the volume of the solid formed by revolving the region bounded by y =  x and y = x² over the interval [0, 1] about the x – axis.

The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk Example of rotating the region about y-axis

Example: rotate it around x = axis

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

Find the volume of the solid generated by revolving the regions about the x-axis. bounded by

Find the volume of the solid generated by revolving the regions about the x-axis.bounded by

Find the volume of the solid generated by revolving the regions about the y-axis. bounded by

Find the volume of the solid generated by revolving the regions about the line y = -1.bounded by

The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius Example of a washer

If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r

The volume of the solid generated by revolving the first quadrant region bounded by the curve and the lines x = ln 3 and y = 1 about the x-axis is closest to a) 2.79 b) 2.82 c) 2.85 d) 2.88 e) 2.91 CALCULATOR REQUIRED

Volumes by Cylindrical Shells

Shell Method Based on finding volume of cylindrical shells –Add these volumes to get the total volume Dimensions of the shell –Radius of the shell –Thickness of the shell –Height

The Shell Consider the shell as one of many of a solid of revolution The volume of the solid made of the sum of the shells f(x) g(x) x f(x) – g(x) dx

Hints for Shell Method Sketch the graph over the limits of integration Draw a typical shell parallel to the axis of revolution Determine radius, height, thickness of shell Volume of typical shell Use integration formula

Consider the region bounded by x = 0, y = 0, and

Rotation About x-Axis Rotate the region bounded by y = 4x and y = x 2 about the x-axis What are the dimensions needed? –radius –height –thickness radius = y thickness = dy

Rotation About Non-coordinate Axis Possible to rotate a region around any line Rely on the basic concept behind the shell method x = a f(x) g(x)

Rotation About Non-coordinate Axis What is the radius? What is the height? What are the limits? The integral: x = a f(x) g(x) a – x f(x) – g(x) x = c r c < x < a

Rotate the region bounded by 4 – x 2, x = 0 and, y = 0 about the line x = 2 Determine radius, height, limits 4 – x 2 r = 2 - x

u = x – 1 x = 1 u=0 x = u + 1 x = 2 u=1 du = dx

Blobs in Space Volume of a blob: Cross sectional area at height h: A(h) Volume =

Volumes with cross-sections: We will be given a “boundary” for the base of the shape which will be used to find a length. We will use that length to find the area of a figure generated from the slice. The dy or dx will be used to represent the thickness. The volumes from the slices will be added together to get the total volume of the figure.

Procedure: volume by slicing o sketch 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

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square. Bounds? Top Function? Bottom Function? [-1,1] Length?

We use this length to find the area of the square. Length? Area? Volume?

Using the half circle [0,1] as the base slices perpendicular to the x-axis are isosceles right triangles. Length? Area? Volume? Bounds? [0,1]

Find the volume of a solid whose base is the circle x 2 + y 2 = 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 length of a side is :

Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the circle.

The base of the solid is the region between the curve and the interval [0,π] on the x-axis. The cross sections perpendicular to the x-axis are equilateral triangles with bases running from the x-axis to the curve. Bounds: Top Function: Bottom Function: [0,π] Length: Area of an equilateral triangle:

Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all equilateral triangles whose sides lie on the base of the circle.

Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all semicircles whose sides lie on the base of the circle.

Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all Isosceles right triangles whose sides lie on the base of the circle.

Let R be the region marked in the first quadrant enclosed by the y-axis and the graphs of as shown in the figure below R a)Setup but do not evaluate the integral representing the volume of the solid generated when R is revolved around the x-axis. b)Setup, but do not evaluate the integral representing the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares.

CALCULATOR REQUIRED

NO CALCULATOR

Let R be the region in the first quadrant under the graph of a) Find the area of R. b)The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? c)Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares. CALCULATOR REQUIRED

Let R be the region in the first quadrant under the graph of a) Find the area of R.

Let R be the region in the first quadrant under the graph of b)The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? A

Let R be the region in the first quadrant under the graph of c)Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

Let R be the region in the first quadrant bounded above by the graph of f(x) = 3 cos x and below by the graph of a)Setup, but do not evaluate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the x-axis. b)Let the base of a solid be the region R. If all cross sections perpendicular to the x-axis are equilateral triangles, setup, but do not evaluate, an integral expression of a single variable for the volume of the solid.