Application of integration

G.K. BHARAD INSTITUTE OF ENGINEERING Prepared by :- (1) Shingala nital (2) Paghdal Radhika (3) Bopaliya Mamata

 Defination of volume  Volume by Slicing  Volume of solid of revolution Washer method  Volume of solid of revolution Disk method  Volume by Cylindrical Shell

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 solid 1. 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.

Y X 0 S Px a x b Cross-section R(x) With area A(x)

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

Y X

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 ]

A(x) dx

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).

h r1 r2

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 = ∫ 2x f(x) dx Volume = (curcumference)(height)(thickness) V= (2r) h ∂r Rotation about x- axis Y = ∫ 2y f(y) dy