INSTRUCTORS Raffi Hovasapian John Zhu

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 0 answersPost by Steve Denton on October 22, 2012At 21:30 in example 4, how exactly did the 1/2 cancel with the division by 2 in the next term? 1 answerLast reply by: Jingwei XieMon Apr 21, 2014 10:06 PMPost by Steve Denton on October 22, 2012At 14:40 or so, isn't the height of an isosceles triangle s(root3)/2 and the area then, not s^2, but s^2(root3)/4?

Revolving Solids Known Cross Sections

• Finding volume of solid with known cross sections and bounds
• Find area of known cross-section
• Substitute bounding functions to variables in area equation appropriately
• Evaluate integral of area of known cross-section
• Knowing derivations of disk, washer, and cylindrical formulas dramatically improves understanding of known cross-sections volumes

Revolving Solids Known Cross Sections

Find the volume of a solid whose base is defined by the region bounded by y = x, y = 0, and x = 1 with the diameters of semi-circle cross-sections along the y-axis.
• Area of a semi-circle = [1/2] πr2
• V = [1/2] π∫01 ([x/2])2 dx
• V = [(π)/8] [(x3)/3] |01
V = [(π)/24]
Find the volume of a solid whose base is defined by the region bounded by y = sinx and y = 0 from x = 0 to x = π with the square cross-sections.
• A = l2
• V = ∫0π y2 dx
• V = ∫0π sin2 x  dx
• V = ∫0π ([(1 − cos2x)/2]) dx
• V = [1/2] ∫0π 1 − cos2x  dx
• V = [1/2] (∫0π dx − ∫0π cos2x dx)
• u = 2x
• du = 2  dx
• V = [1/2] (x |0π − [1/2] sin2x |0π)
V = [(π)/2]
Find the volume of a solid whose base is defined by the region bounded by y = x2 and y = x with equilateral triangle cross-sections along the y-axis.
• Area of equilateral triangle = [(√3)/4] l2
• Intersection points
• x2 = x
• x = 0, 1
• In between 0 and 1, x ≥ x2
• l = x − x2
• V = [(√3)/4] ∫01 (x − x2)2 dx
• V = [(√3)/4] ∫01 x4 − 2x3 + x2  dx
• V = [(√3)/4] ([(x5)/5] − [(x4)/2] + [(x3)/3]) |01
• V = [(√3)/4] ([1/5] − [1/2] + [1/3] − 0)
V = [(√3)/120]
Find the volume of a solid whose base is defined by the region bounded by y = x2 and y = x with square cross-sections along the x-axis.
• This time, we're taking triangles parallel to the x-axis
• We need to rewrite both of the equations in terms of x
• x2 = y
• x = ±√y
• The intersection is in the first quadrant, so we only care about the positive value
• x = √y
• Intersects at y = 0, 1
• A = l2
• l = √y − y
• V = ∫01 (√y − y)2 dy
• V = ∫01 (y − 2y√y + y2) dy
• V = ∫01 (y − 2 y[3/2] + y2) dy
• V = [(y2)/2] − [(4 y[5/2])/5] + [(y3)/3] |01
• V = [1/2] − [4/5] + [1/3]
V = [1/30]
Find the volume of a solid whose base is defined by the region bounded by y = 0, y = 2, x = 0, and x = 6 with semi-circle cross-sections along the x-axis.
• The area bound by those lines is simply a square, with lines parallel to the axes.
• A = [1/2] πr2
• r = [6/2]
• V = [(π)/2] ∫02 32 dy
• V = [(π)/2] 9x |02
V = 9π
Find the volume of a solid whose base is defined by the region bounded by y = x2 + 1 and y = −x2 + 3 with semi-circle cross-sections along the y-axis.
• Find their intersection points
• x2 + 1 = −x2 + 3
• 2x2 = 2
• x = −1, 1
• A = [1/2] πr2
• r = [((−x2 + 3) − (x2 + 1))/2]
• V = [(π)/2] ∫−11 (−x2 + 1)2  dx
• V = [(π)/2] ∫−11 x4 − 2x2 + 1  dx
• V = [(π)/2] ([(x5)/5] − [(2x3)/3] + x) |−11
• V = [(π)/2] ([1/5] − [2/3] + 1 − ([(−1)/5] − [(−2)/3] − 1))
• V = [(π)/2] ([1/5] − [2/3] + 1 + [1/5] − [2/3] + 1)
V = [(8π)/15]
Find the volume of a solid whose base is defined by the region bounded by y = cosx and y = 0 from x = 0 to x = [(π)/2] with square cross-sections along the y-axis.
• V = ∫0[(π)/2] cos2 x  dx
• V = ∫0[(π)/2] [(1 + cos2x)/2] dx
• V = [1/2] (x + [1/2] sin2x) |0[(π)/2]
• V = [1/2] ([(π)/2] + 0)
V = [(π)/4]
Find the volume of a solid whose base is defined by the region bounded by y = √{1 − x2} and y = 0 with semi-circle cross-sections along the y-axis.
• V = [(π)/2] ∫−11 ([(√{1 − x2})/2])2 dx
• V = [(π)/8] ∫−11 1 − x2  dx
• We can use symmetry
• V = [(π)/4] ∫01 1 − x2  dx
• V = [(π)/4] (x − [(x3)/3]) |01
• V = [(π)/4] (1 − [1/3] − 0)
V = [(π)/6]
Find the volume of a solid whose base is defined by the region bounded by x = y3 and x = y2 with square cross-sections along the x-axis.
• V = ∫01 (y2 − y3)2 dy
• V = ∫01 y4 − 2 y5 + y6  dy
• V = ([(y5)/5] − [(y6)/3] + [(y7)/7]) |01
• V = [1/5] − [1/3] + [1/7]
V = [1/105]
Find the volume of a solid whose base is defined by the region bounded by x = y3 and x = y2 with square cross-sections along the y-axis.
• Rewrite in terms of y
• y = 3√{x}   and  y = ±√x
• First quadrant again, so we're only interested in the positive
• V = ∫01 (√x − 3√{x})2 dx
• V = ∫01 x − 2 x[5/6] + x[2/3]  dx
• V = ([(x2)/2] − [(12 x[11/6])/11] + [(3 x[5/3])/5]) |01
• V = [1/2] − [12/11] + [3/5]
V = [1/110]

*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.