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For more information, please see full course syllabus of College Calculus: Level I

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### Volume by Method of Disks and Washers

• I highly recommend starting with a sketch of the functions involved and drawing in a sample radius!
• I recommend memorizing the formulas but also understanding the geometry that leads to the formulas.
• If an interval is not given, you may need to set the two functions equal in order to determine the interval involved.
• Sometimes, geometrical considerations can help you to check your results or even provide a non-calculus way to finish a problem.

### Volume by Method of Disks and Washers

Find the volume of the solid that results from revolving y = x from x = 0 to x = 4 around the x-axis.
• The area of a slice would be πr2. The volume would be πr2 * thickness
• So the volume of a tiny slice would be πr2 dx where dx is the thickness
• In this problem, we're rotating around the x-axis, so the radius is just y in this case.
• Volume of slice = πy2 dx = πx2 dx
• Total volume from x = 0 to x = 4 is V
• V = ∫04 πx2 dx
• V = π[(x3)/3] |04
• V = π[64/3] − 0
• We just found the volume of a cone using calculus!
V = [(64 π)/3]
Find the volume of the solid that results from revolving x = 4 from y = 0 to y = 6 around the y-axis.
• V = π∫06 42 dy
• V = π16x |06
• V = 96π− 0
• Now we found the volume of a cylinder.
V = 96π
Find the volume of the solid that results from revolving y = secx from x = −[(π)/4] to x = [(π)/4] around the x-axis.
• V = π∫−[(π)/4][(π)/4] sec2 x  dx
• V = πtanx |−[(π)/4][(π)/4]
• V = π(tan[(π)/4] − tan[(−π)/4])
• V = π(1 − (−1))
V = 2π
Find the volume of the solid that results from revolving y = lnx from y = 0 to y = 1 about the y-axis.
• V = π∫01 x2 dy
• y = lnx
• ey = elnx
• x = ey
• V = π∫01 e2y dy
• u = 2y
• du = 2  dy
• V = [(π)/2] ∫01 eu du
• V = [(π)/2] eu |01
• V = [(π)/2] e2y |01
• V = [(π)/2] (e2 − e0)
V = [(π)/2] (e2 − 1)
Find the volume of the solid that results from revolving y = [lnx/(√x)] from x = 1 to x = e about the x-axis.
• V = π∫1e [((lnx)2)/x] dx
• u = lnx
• du = [1/x] dx
• V = π∫1e u2 du
• V = π[(u3)/3] |1e
• V = π[((lnx)3)/3] |1e
• V = [(π)/3] ( (lne)3 − (ln1)3 )
V = [(π)/3]
Find the volume of the solid that results from revolving The area bounded by y = ex, y = 1, and x = 2 around the x-axis.
• V = π∫02 (ex)2 − 12 dx
• V = π∫02 e2x − 1  dx
• V = π([1/2] e2x − x) |02
• V = π([1/2] e4 − 2 − [1/2] e0 + 0)
• V = π([1/2] e4 − [5/2])
V = [(π)/2] (e4 − 5)
Find the volume of the solid that results from revolving y = x2 + 1 from x = 0 to x = 1 around the x-axis.
• V = π∫01 (x2 + 1)2 dx
• V = π∫01 x4 + 2x2 + 1  dx
• V = π([(x5)/5] + [(2x3)/3] + x) |01
• V = π([1/5] + [2/3] + 1 − 0 − 0 − 0)
V = [(28π)/15]
Find the volume of the solid that results from revolving y = √{4 − x2} from x = −2 to x = 2 around the x-axis.
• V = π∫−22 (√{4 − x2})2 dx
• V = π∫−22 4 − x2 dx
• V = π(4x − [(x3)/3]) |−22
• V = π(8 − [8/3] − (−8 + [8/3]))
• V = π(16 − [16/3])
• This is the volume of a sphere of radius 2!
V = [(32π)/3]
Find the volume of the solid that results from revolving y = [1/(√x)] from x = 1 to x = 3 around the x-axis.
• V = π∫13 [1/x] dx
• V = πlnx |13
• V = π(ln3 − ln1)
• V = π(ln3 − 0)
V = πln3
The area bounded by y = x3 and y = x is rotated about the x-axis. Find the volume of the resulting solid.
• Find their points of intersection
• x3 = x
• x = 0, 1, −1
• V = π(− ∫−10 (x)2 − (x3)2 dx) + π∫01 (x)2 − (x3)2 dx
• We can use symmetry here instead of using two separate integrals
• V = 2 π∫01 (x)2 − (x3)2 dx
• V = 2 π([(x3)/3] − [(x7)/7]) |01
• V = 2π([1/3] − [1/7] − 0)
V = [(8π)/21]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Volume by Method of Disks and Washers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Important Equations 0:16
• Equation 1: Rotation about x-axis (disks)
• Equation 2: Two curves about x-axis (washers)
• Equation 3: Rotation about y-axis
• Lecture Example 1 6:05
• Lecture Example 2 8:28
• Lecture Example 3 11:55
• Additional Example 4
• Additional Example 5