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Application of Integrals

• Area of the region bounded by the curve y = f(x), x-axis, and the lines x = a and x = b (b > a) is given by  or
• The area of the region bounded by the curve x = g(y), y-axis, and the lines y = c and y = d is given by  or
• If a line y = mx + p intersects a curve y = f(x) at x = a and x = b, (b > a), then the area (A) of region bounded by the curve y = f(x) and the line y = mx + p is

• If a line y = mx + p intersects a curve x = g(y) at y = c and y = d ,(d > c), then the area (A) of region bounded by the curve x = g(y) and the line y = mx + p is

Example 1: Find the area of the region in the first and third quadrant enclosed by the x-axis and the line , and the ellipse

Solution: The given equations are

... (1)

... (2)

Substituting  in equation (2), we obtain

Hence, the line meets the ellipse at C  and D  in the first and third quadrant respectively.

In the figure, CM ⊥ XX′

Now, area OCMO =

Area ACMA

1. The area of the region enclosed bet…

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