Volume: The Disc Method Section 6.2

If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disc, which is formed by revolving a rectangle about an axis .

Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve about the x axis What kind of functions generated these solids of revolution? f(x) a b

Disks f(x) We seek ways of using integrals to determine the volume of these solids Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume? dx

If a region of a plane is revolved about a line, the resulting solid is the solid of revolution, and the line is the axis of revolution. The simplest is a right circular cylinder or disk, a rectangle revolved around the x-axis. The volume is equal to the area of the disk times the width of the disk, V = πr2w. r w

The Disc Method Volume of disc = (area of disc)(width of disc)

Disks To find the volume of the whole solid we sum the volumes of the disks Shown as a definite integral f(x) a b

w w R R Axis of Revolution

Revolve this rectangle about the x-axis. Revolve this function about the x-axis.

Revolve this rectangle about the x-axis. Forms a Disk. Revolve this function about the x-axis.

Revolving About y-Axis Also possible to revolve a function about the y-axis Make a disk or a washer to be horizontal Consider revolving a parabola about the y-axis How to represent the radius? What is the thickness of the disk?

Revolving About y-Axis Must consider curve as x = f(y) Radius = f(y) Slice is dy thick Volume of the solid rotated about y-axis

Horizontal Axis of Revolution Volume = V = Vertical Axis of Revolution

Disk Method (to find the volume of a solid of revolution) Horizontal Axis of revolution: Vertical Axis of revolution:

Disk Method (to find the volume of a solid of revolution) Horizontal Axis of revolution: radius width Vertical Axis of revolution:

1. Find the volume of the solid formed by 1. Find the volume of the solid formed by revolving f(x) over about the x-axis.

1. Find the volume of the solid formed by 1. Find the volume of the solid formed by revolving f(x) over about the x-axis.

2. Find the volume of the solid formed by revolving the 2. Find the volume of the solid formed by revolving the region formed by f(x) and g(x) about y = 1.

2. Find the volume of the solid formed by revolving the 2. Find the volume of the solid formed by revolving the region formed by f(x) and g(x) about y = 1. Length of Rectangle:

3. Find the volume of the solid formed by 3. Find the volume of the solid formed by revolving the region formed by over about the y-axis.

3. Find the volume of the solid formed by 3. Find the volume of the solid formed by revolving the region formed by over about the y-axis.

The disc method can be extended to cover solids of revolution with holes by replacing the representative disc with a representative washer. w R r

Washers Consider the area between two functions rotated about the axis Now we have a hollow solid We will sum the volumes of washers As an integral f(x) g(x) a b

Volume of washer =

How do you find the volume of the figure formed by revolving the shaded area about the x-axis?

How do you find the volume of the figure formed by revolving the shaded area about the x-axis? Outside radius Inside radius The volume we want is the difference between the two. Revolving this function creates a solid whose volume is larger than we want. Revolving this function carves out the part we don’t want.

Washer Method (for finding the volume of a solid of revolution)

Washer Method (for finding the volume of a solid of revolution) Outside radius Inside radius

1. Find the volume of the solid generated by revolving 1. Find the volume of the solid generated by revolving the area enclosed by the two functions about the x- axis.

1. Find the volume of the solid generated by revolving 1. Find the volume of the solid generated by revolving the area enclosed by the two functions about the x- axis. Find intersection points first. Washer Method:

2. Find the volume of the solid formed by revolving 2. Find the volume of the solid formed by revolving the area enclosed by the given functions about the y- axis.

2. Find the volume of the solid formed by revolving 2. Find the volume of the solid formed by revolving the area enclosed by the given functions about the y- axis. This region is not always created by the same two functions. The change occurs at y = 1. We need to use two integrals to find the volume. For y in [0,1] we use disk method. For y in [1,2] we use washer method. Since we’re revolving about the y-axis, each radius must be in terms of y. (distance from the parabola to the x-axis) (distance from the parabola to the y-axis)

Volume of Revolution - X Find the volume of revolution about the x-axis of f(x) = sin(x) + 2 from x = 0 to x = 2.

Volume of Revolution - X Find the volume of revolution about the x-axis of f(x) = sin(x) + 2 from x = 0 to x = 2 Use the TI to integrate this one! Did you get 88.826 cubic units?

Volumes of Solids with Known Cross Sections For cross sections of area A(x) taken perpendicular to the x-axis, Volume = For cross sections of area A(y) taken perpendicular to the y-axis, Volume =

Volume of Revolution - X Let’s look at the cross section or slice. What is it? It’s a circle. What is the radius? f(x). What is the area? A = r2 A =  (f(x))2

Volume of Revolution - X How wide (thick) is the disc? dx The volume of the disk is V =  (f(x))2dx How do we add up all the disks from x = 1 to x = 4?

Example 1 Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares; (b) semicircles; and (c) equilateral triangles.

Example 1 (a) Square Cross Sections My Advice??? Draw the 2-D picture and imagine the cross-sectional shape coming out in the third dimension! Try and think of how to write the area of these cross sections using the given function. Set up the integral and integrate. Think: SKATE BOARD RAMP!!!

Example 1 (b) Semicircular Cross Sections Think: CORNICOPIA!!!

Example 1 (c) Equilateral Triangular Cross Sections Think: PYRAMID, kinda??

Example 2 Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares; (b) semicircles; and (c) equilateral triangles.

Example 2 (a) Square Cross Sections Think: PYRAMID!!!

Example 2 (b) Semicircular Cross Sections Think: CORNICOPIA AGAIN!!!

Example 2 (c) Equilateral Triangular Cross Sections Think: PYRAMID!!!

Example 2 (c) Equilateral Triangular Cross Sections Think: PYRAMID!!!

Theorem:. The volume of a solid with cross-section of area Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by a b Finding the volume of a solid with known cross-section is a 3-step process:

Theorem:. The volume of a solid with cross-section of area Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by ***This circle is the base of a 3-D figure coming out of the screen. S a b ***This rectangle is a side of a geometric figure (a cross-section of the whole). Finding the volume of a solid with known cross-section is a 3-step process: Step #1: Find the length (S) of the rectangle used to create the base of the figure. Step #2: Find the area A(x) of each cross-section (in terms of this rectangle). Step #3: Integrate the area function from the lower to the upper bound. Volume of a solid with cross-sections of area A(y) and perpendicular to the y-axis: d S c

Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are a. squares Step #1: Find S. Step #2: Find A(x). Step #3: Integrate the area function.

Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are a. squares Step #1: Find S. S -2 2 Step #2: Find A(x). S Step #3: Integrate the area function.

b. equilateral triangles Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are b. equilateral triangles Step #1: Find S. Step #2: Find A(x). Step #3: Integrate.

b. equilateral triangles Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are b. equilateral triangles Step #1: Find S. S -2 2 Step #2: Find A(x). S Step #3: Integrate.

Solids with known cross sections Find the volume of a solid where the base is an ellipse with semi-major axis of 4 units and semi-minor axis of 3 units if all cross sections perpendicular to the major axis are semicircles. Volumes by Slicing x 4 3 y 2y y x

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.