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CALCULUSApplication of
integration
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G.K. BHARAD INSTITUTE OF ENGINEERING
Prepared by :- (1) Shingala nital (2) Paghdal Radhika (3) Bopaliya Mamata
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Cantents
Defination of volume Volume by Slicing Volume of solid of revolution Washer
method Volume of solid of revolution Disk method Volume by Cylindrical Shell
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Defination of Volume
Let S be a solid that lies between x=a and x=b. If the cross-sectional area of S in the plane Px, through x and perpendicular to the x – axis , is A(x) , where A is a continuous function , then the volume of S is
V = ∫ A(x) dx = A(b – a)Procedure for calculating the volume of a solid1. Sketch the solid with typical cross section.2. Find a formula for A(x), the area of a typical cross
section.3. Find the limits of integration.4. Integrate A(x) using the formula.
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Volume by Slicing
Y
X
0
S
Px
a x
b
Cross-section R(x)With area A(x)
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Volume by slicing
A cross-section of the solid S formed by intersecting S(solid) with a plane Px perpendicular to the x-axis through the point x in the interval [a, b]
The volume of cylindrical solid is always defined to be its base area times its height.
The volume of the cylindrical solid is VOLUME = AREA * HEIGHT = A.h
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Volume of solid of revolution Washer method
Y
X
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Volume of solid of revolution Washer method
In a washer method a slab is a circular washer of outer radius R(x) and inner radius r(x), hence
A(x) = [R(x)]2 – r(x)2]
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Volume of solid of revolution Disk method
A(x)
dx
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Volume of solid of revolution Disk method
The solid generated by rotating a plane region about an axis in its plane is called solid of revolution.
find the cross sectional area A(x) of a disk of radius R(x).The area is then
A(x) = (radius)2 = [R(x)]2
So the volume is V = ∫ A(x) dx = ∫ [R(x)]2 dx This method is called the disk method because
a cross section is a circular disk of radius R(x).
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Volume by Cylindrical Shell
h
r1
r2
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Volume by Cylindrical Shell
Some volume problems are very difficult to handle by the method of preceding section. Fortunetly, there is a method, called the method of cylindrical shell.
Rotation about y-axis X = ∫ 2x f(x) dx
Volume = (curcumference)(height)(thickness) V= (2r) h ∂r
Rotation about x- axis Y = ∫ 2y f(y) dy
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