Z b Z b Disk/Washer Method x dx y dyevergreen.loyola.edu/loberbroeckling/www/ma252/volumes.pdfMA 252...
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MA 252 Volumes of Solids of Revolution 1 Disk/Washer Method b a A(x) dx or b a A(y) dy Take cross-sections PERPENDICULAR to axis of revolution. If cross-section is a solid disk, A = πR 2 If cross-section is a washer/ring/annulus, A = πR 2 - πr 2 Axis of Revolution is HORIZONTAL: integrate with respect to x: a b R f x a b V = b a π[f (x)] 2 dx a b R f x r gx a b V = b a π[f (x)] 2 - π[g(x)] 2 dx a b R f x c r c c a b V = b a π[f (x)+ c] 2 - πc 2 dx a b R f x c r gx c c a b V = b a π[f (x)+ c] 2 - π[g(x)+ c] 2 dx a b r c f x R c c a b V = b a πc 2 - π[c - f (x)] 2 dx a b r c f x c R c gx a b V = b a π[c - g(x)] 2 - π[c - f (x)] 2 dx Examples of regions that can be done with either the disk/washer method or the shell method: see §6.2, #19-30.
Transcript of Z b Z b Disk/Washer Method x dx y dyevergreen.loyola.edu/loberbroeckling/www/ma252/volumes.pdfMA 252...
MA 252 Volumes of Solids of Revolution 1
Disk/Washer Method∫ b
a
A(x) dx or∫ b
a
A(y) dy
Take cross-sections PERPENDICULAR to axis of revolution.If cross-section is a solid disk, A = πR2
If cross-section is a washer/ring/annulus, A = πR2 − πr2
Axis of Revolution is HORIZONTAL: integrate with respect to x:
a b
R = f HxL
a b
V =∫ b
a
π[f(x)]2 dx
a b
R = f HxL
r = gHxL
a b
V =∫ b
a
π[f(x)]2 − π[g(x)]2 dx
a b
R = f HxL + c
r = c-c
a b
V =∫ b
a
π[f(x) + c]2 − πc2 dx
a b
R = f HxL + c
r = gHxL + c
-c
a b
V =∫ b
a
π[f(x) + c]2 − π[g(x) + c]2 dx
a b
r = c - f HxL
R = c
c
a b
V =∫ b
a
πc2 − π[c− f(x)]2 dx
a b
r = c - f HxL
c
R = c - gHxL
a b
V =∫ b
a
π[c− g(x)]2 − π[c− f(x)]2 dx
Examples of regions that can be done with either the disk/washer method or the shell method:see §6.2, #19-30.
MA 252 Volumes of Solids of Revolution 2
Disk/Washer Method (cont.)∫ b
a
A(x) dx or∫ b
a
A(y) dy
Take cross-sections PERPENDICULAR to axis of revolution.If cross-section is a solid disk, A = πR2
If cross-section is a washer/ring/annulus, A = πR2 − πr2
Axis of Revolution is VERTICAL: integrate with respect to y:
a
b
R = f HyL
a
b
V =∫ b
a
π[f(y)]2 dy
a
b
R = f HyL
r = gHyL
a
b
V =∫ b
a
π[f(y)]2 − π[g(y)]2 dy
a
b
-c
R = f HyL + cr = c
a
b
V =∫ b
a
π[f(y) + c]2 − πc2 dy
a
b
-c
R = f HyL + cr = gHyL + c
a
b
V =∫ b
a
π[f(y) + c]2 − π[g(y) + c]2 dy
a
b
c
R = c
r = c - f HyL
a
b
V =∫ b
a
πc2 − π[c− f(y)]2 dy
a
b
c
R = c - gHyL
r = c - f HyL
a
b
V =∫ b
a
π[c− g(y)]2 − π[c− f(y)]2 dy
Examples of regions that are best to use the disk/washer method:y = 1/x, x = 1, x = 2, y = 0 about the x-axis, or about the lines y = −1, y = 5y = cos x, y = sinx, x = 0, x = π/6 about the x-axis, or about the lines y = 1, y = −1x = 2y − y2, x = 0, about the y-axis, or about the lines x = 5, x = −5
MA 252 Volumes of Solids of Revolution 3
Shell Method∫ b
a
2πRh dx or∫ b
a
2πRh dy
Take cross-sections PARALLEL to axis of revolution.Figure out the radius R from cross-section to the axis of revolutionFigure out the height h of the cross-section
Axis of Revolution is VERTICAL: integrate with respect to x:
a b
h = f HxL
R = x
a b
V =∫ b
a
2πxf(x) dx
a b
h = f HxL - gHxL
R = x
a b
V =∫ b
a
2πx[f(x)− g(x)] dx
a b
-c
R = x + c
h = f HxL
a b
V =∫ b
a
2π(x + c)f(x) dx
a b
-c
R = x + c
h = f HxL - gHxL
a b
V =∫ b
a
2π(x + c)(f(x)− g(x)) dx
a b
c
R = c - x
h = f HxL
a b
V =∫ b
a
2π(c− x)f(x) dx
a b
c
R = c - x
h = f HxL - gHxL
a b
V =∫ b
a
2π(c− x)(f(x)− g(x)) dx
Examples of regions that are best to use the shell method:y = 1/x, x = 1, x = 2, y = 0 about the y-axis, or about the lines x = 3, x = 0.5y = cos x, y = sinx, x = 0, x = π/6 about the y-axis, or about the lines x = 2, x = −2x = 2y − y2, x = 0, about the x-axis, or about the lines y = 5, y = −5
MA 252 Volumes of Solids of Revolution 4
Shell Method (cont.)∫ b
a
2πRh dx or∫ b
a
2πRh dy
Take cross-sections PARALLEL to axis of revolution.Figure out the radius R from cross-section to the axis of revolutionFigure out the height h of the cross-section
Axis of Revolution is HORIZONTAL: integrate with respect to y:
a
b
h = f HyL
R = y
a
b
V =∫ b
a
2πyf(y) dy
a
b
h = f HyL - gHyL
R = ya
b
V =∫ b
a
2πy[f(y)− g(y)] dy
a
b
-c
h = f HyL
R = y + ca
b
V =∫ b
a
2π(y + c)f(y) dy
a
b
-c
h = f HyL - gHyL
R = y + c
a
b
V =∫ b
a
2π(y + c)(f(y)− g(y)) dy
a
b
c
h = f HyL
R = c - y
a
b
V =∫ b
a
2π(c− y)f(y) dy
a
b
c
h = f HyL - gHyL
R = c - y
a
b
V =∫ b
a
2π(c− y)(f(y)− g(y)) dy
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