Application of Integrals Class 12 Notes Mathematics Chapter 8

Chapter at a Glance

1. Area Under Simple Curves

(i) Let us find the area bounded by the curve y = f (x), x-axis and the ordinate x = a and x = b. Consider the area under the curve as composed of large number of thin vertical strips. Let us consider an arbitrary strip of height y and width dx. Area of this strip dA= ydx, where y = f (x). Total Area A of the region between x-axis, ordinates x = a, x = b and the curve y = f (x) is sum of areas of the thin strips across the region PQMLP

(ii) To find the Area of bounded Regions, we may use the following Algorithm.

Step. 1 Draw a rough sketch showing the region whose area is to be found.

Step. 2 Slice the area into horizontal or vertical strips as the case may be.

Step. 3 Consider an arbitrary strip approximatly as a rectangle.

Step. 4 Find the area of rectangle. If the rectangular strip is parallel to y-axis then its width is taken as dx and if it is parallel to x-axis, then its width is taken as dy. In given fig.1, rectangle RLMQ has area = y. dx, and in fig (2) the area of the rectangle RLMQ = x.dy.

Step. 5 Find the limit within which the rectangle can move.

In fig (1), the rectangle of area y.dx can move between x = a and x = b, therefore the area of the region bounded by y = f(x), y = 0, x = a and x = b is given by.

In fig (2), the rectangle of area x.dy can move between y = c and y = d, Therefore area of the region bounded by x = g (y), x = 0, y = c and y = d is given by dy

2. Area between Two curves


(iii) Ellipse:

represents an ellipse with center at origin and major axis is x-axis, minor axis is y-axis. Length of major axis is 2a and length of minor axis is 2b. Ellipse intersect the x-axis at a and –a and intersects the y-axis at b and –b.

v) Graph of y = sin x

(vi) Graph of y = cos x

Related Articles:

Share this: