CALCULUSApplication of

integration

G.K. BHARAD INSTITUTE OF ENGINEERING

Prepared by :- (1) Shingala nital (2) Paghdal Radhika (3) Bopaliya Mamata

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

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.

Volume by Slicing

Y

X

0

S

Px

a x

b

Cross-section R(x)With area A(x)

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

Volume of solid of revolution Washer method

Y

X

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]

Volume of solid of revolution Disk method

A(x)

dx

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).

Volume by Cylindrical Shell

h

r1

r2

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