### 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**

**Note: **

**(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**

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