6.3 Volume by Slices Thurs April 9 Do Now Evaluate each integral 1) 2)

HW Review: worksheet480 # ) 621a) 263/4 = ) b) 31 5) 2/pi =.63722a) 3.5 6).95522b) ).582 8) 1.775

The Volume Problem What objects do we know the volume of? What kind of objects don’t we know the volume of?

Volumes by Slicing Let’s place a volume on the x-axis on its side. Then the x value becomes the height By slicing the volume into prisms, we get

Solids of Revolution Suppose f(x) is greater than 0 and continuous on the interval [a, b]. We can create a solid of revolution by revolving f(x) around the x-axis. Common solids of revolutions are cones, and cylinders

Volumes by Disks Suppose f(x) > 0 and continuous on the interval [a,b], and a solid of revolution is created by revolving around the x-axis. Then the volume of the solid is

Volumes by Disks 1) Draw the region 2) Determine the radius (f(x)) 3) Determine the bounds (lowest and highest x values) 4) Integrate the volume using disks

Ex 2.3 Suppose a region is bounded by x = 0, x = 12, y = 0, and y = 1/3x + 1, and is revolved about the x-axis. Find the volume of this solid

Ex 2.4 Revolve the region under the curve on the interval [0,4] about the x-axis and find the volume of this solid.

Closure Hand in: Find the volume of the region bounded by y = x + 1, y = 0, and x = 3, and revolved about the x-axis HW: p.381 #1-11 odds, 33

Method of Disks Using the Y Variable Fri April 10 Do Now Revolve the region under the curve bounded by y = 0, x = 0, and x = 4 about the x-axis, and find the volume of this solid.

HW Review: p.381 # ) 21pi = ) 21/2 pi = 10.5pi = ) 81/10 pi = 8.1pi = ) 24573/13 pi = pi = ) pi = ) 3.19pi = ) 376/15 pi = pi = 78.75

Using Y as the Independent Variable Suppose g(y) > 0 and g is continuous on the interval [c,d]. Then, revolving the region bounded by the curve x = g(y) about the y-axis generates a solid whose volume is

X or Y as our variable of integration? The decision on which variable to integrate with depends on the axis with which we are revolving the solid around. Revolve around x-axis (horizontal line) –Integrate with respect to x (y = f(x)) Revolve around y-axis (vertical line) –Integrate with respect to y (x = g(y))

Revolving About the Y-axis 1) Draw the region 2) Rewrite the function in terms of y 3) Determine the Y-bounds 4) Integrate the volume using disks

Ex 2.5 Find the volume of the solid resulting from revolving the portion of the curve from x = 0 to x = 2 about the y-axis

Ex 2.6a Let R be the region bounded by the graphs of, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the y-axis

Closure Journal Entry: Describe how we would find the volume of a solid revolved about the x-axis. What is the difference if we revolve about the y-axis? HW: p.381 #

6.3 Volumes by Washers Mon April 13 Do Now Find the volume of each region 1) Bounded by, y = 0 and x = 2 revolved about the x-axis 2) Bounded by, x = 0 and y = 4 revolved about the y-axis

HW Review: p.381 # ) 15/2 pi = 7.5pi = ) 32pi = ) 8pi = ) 250pi = ) 32/35 pi =.914pi = 2.871

Volumes with space between the region and the axis When there is a space between the region and axis of revolution, we cannot use disks. We need to subtract the extra space.

Washer Method Suppose f(x) is continuous on the interval [a,b] and does not cross the axis of revolution, but creates a cavity inside the solid when revolved, then

Washer Method 1) Draw the region 2) Determine axis of revolution and rewrite functions if necessary (x or y) 3) Determine the two radii of the solid 4) Determine bounds (x or y) 5) Set up integral using washer method 6) Integrate

Ex 2.6b Let R be the region bounded by the graphs of, x = 0 and y = 1. Compute the volume of the solid formed by revolving R about the x-axis.

Ex Let R be the region bounded by the graphs y = x^2, x = 0, and y = 8. Find the volume of the solid formed by revolving R about the x-axis

Closure Hand in: Let R be the region bounded by the graphs, x = 0, and y = 2. Find the volume of the solid formed by revolving R about the x-axis HW: p.381 #

5.2 Washer + Disk Practice Tues April 14 Do Now Let R be the region bounded by y = 2x, y = 0, and x = 6. Find the volume of the solid formed by revolving R about the y- axis

HW Review: p.381 # ) 256pi = ) 656/3 pi = pi = ) 3/10 pi = 0.3pi = ) 704/15 pi = 46.93pi = ) 8pi + 8pi = 16pi = ) 1400/3 pi = pi =

Practice Worksheet(blue) p.456 #1-4

Closure Journal Entry: Describe how to find a volume using the washer method. What is the difference between the washer method and the disk method? HW: Finish worksheet p.456 #1-4

5.2 Washer + Disk Practice Wed April 15 Do Now Find the volume of the solid generated when the region enclosed by y = x^2 y = 2, and x = 0 is revolved about the y- axis

HW Review: Worksheet p.456 #1-4 1) 8pi = ) 38pi/15 = ) 13pi/6 = ) 9pi/2 =

Worksheet Practice Worksheet (blue) p #7-13 odds, 19, 20, 23, 25

Closure Journal Entry: Explain the difference between rotating about the y-axis, and rotating about the x-axis. Does the solid look different? How? HW: Finish worksheet p #7-13 odds, 19, 20, 23, 25

6.3 Revolving a Region about Different Lines Thurs April 16 Do Now Find the volume of the solid bounded by the functions y = x, x = 2, and y = 0, revolved about the y-axis

HW Review: Worksheet p.456 #7-13 odds 19, 20, 23, 25 7) =.920 9) 256pi/3 = ) 2048pi/15 = ) 4pi = ) 8pi = ) 8pi = ) 72pi/5 = ) =

Revolving about other lines When revolve about a different line, we want to consider if the line is horizontal or vertical –Horizontal: use x bounds –Vertical: use y bounds

Radii To calculate the radii, subtract the function from the center

Revolving a Region about Different Lines 1) Draw the picture 2) Identify whether the line is horizontal (use x) or vertical (use y) 3) Identify the outer radius –The curve furthest away from revolving line –Distance between radius and axis of revolution –(function - center) 4) Identify the inner radius –The curve closer to revolving line –Distance between radius and axis of revolution –(function - center) 5) Identify the bounds (x or y) 6) Use washers or disks to find the volume

Ex 2.6c Let R be the region bounded by the graphs of, x = 0 and y = 1. Compute the volume of the solid formed by revolving R about the line y = 2.

Closure Hand in:Let R be the region bounded by the graphs of and y = 0. Compute the volume of the solid formed by revolving R about the line y = 7. HW: p.381 #

6.3 Revolving about Different Lines Fri April 17 Do Now Find the volume of the solid generated by revolving the region bounded by about the x-axis

HW Review: p.381 # ) (iv) 29) 128/5 pi = ) 40pi = ) 776/15 pi = ) 824/15 pi = ) 1872/5 pi = ) 96/5 pi =

Revolving a Region about Different Lines 1) Draw the picture 2) Identify whether the line is horizontal (use x) or vertical (use y) 3) Identify the outer radius –The curve furthest away from revolving line –Distance between radius and axis of revolution –(top curve - bottom curve) 4) Identify the inner radius –The curve closer to revolving line –Distance between radius and axis of revolution –(top curve - bottom curve) 5) Identify the bounds (x or y) 6) Use washers or disks to find the volume

Things to Note The outer radius is the function furthest away from the axis of revolution (center) –Distance between outer radius and axis of revolution The inner radius is the function closer to the axis of revolution (center) –Distance between inner radius and axis of revolution –If the inner radius is the axis of revolution, then you use the disk method –If the inner radius is another function, then use the washer method

You try Let R be the region bounded by the graphs of and y = 0. Compute the volume of the solid formed by revolving R about the line y = -3.

Practice Worksheet(green) p.424 #21-24

Closure Journal Entry: Draw an example of a solid that would need the disk method. Draw an example of a solid that would need the washer method. What is the difference? HW: worksheet p.424 # Quiz Thurs April 23

6.3 Revolving About Different Lines Mon April 20 Do Now Find the volume of the solid generated by revolving the region bounded by y = 2, y = 2sinx, x = 0, and x = pi/2, about the line y = 2

HW Review worksheet p a) 8pi/3 (8.38)b) 28pi/3 (29.32) 22a) 32pi/5 (20.12) b) 224pi/15 (46.91) 23a) 32pi/5 (20.12) b) 224pi/15 (46.91) 24a) 2pi/3 (2.09)b) 2pi/3 (2.09)

Reminder When revolving about a horizontal line, use x When revolving about a vertical line, use y When revolving about a line, subtract that value from each radius

Practice (green) worksheet P.424 #28ab, 29cdef

Closure Hand in: Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = 1 HW: P.424 #26ab 28ab, 29cdef Quiz Thurs April 23

Review Tues April 21 Do Now Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = -1

HW Review: p.424 #26ab, 28ab 26a) = f) 36pi = b)= a) pi/10 = b) 9pi/14 = c) 18pi = d) 36pi = e) 18pi =

Quiz Review Average Value –Divide integral by length of interval Volumes –Disk Method –Washer Method –Revolving about x-axis –Revolving about y-axis –Revolving about other lines

More Practice P.424 #30acd, 31cdef

HW Review: p.424 #30acd, 31cdef 30a) 512pi/15 = c) 384pi/5 = d) 1408pi/15 = c) pi/6 = d) 7pi/15 = e) 7pi/6 = f) 13pi/15 = 2.723

Closure Journal Entry: When finding volumes, how do we know whether to integrate with respect to x or y? Do we have a choice? Quiz Thurs

Quiz Review Wed April 22 Do Now Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = 1

HW Review: Worksheet (oj)p.407 #7-18 7) 2pi/315) 2pi/3 8) 6pi16) pi 9) 4 - pi17) 117pi/5 11) 32pi/518) 108pi/5 12) 128pi/7 13) 36pi 14) pi/30