AOB is the positive quadrant of the ellipse x 2 a 2 + y 2 b 2 = 1 where OA = a , OB = b . Then find the area between the arc AB and the chord AB of the ellipse.

First draw figure, Now, Area of shaded part = 1 4 × area of ellipse - area of triangle △ A O B A = 1 4 πab - 1 2 ab = 1 4 ab π - 2

The area bounded by the y - axis; y = cos x , y = sin x when 0 ≤ x ≤ π 2 is given by

First draw graph, So, Area = A = ∫ 0 π 4 cos x - sin x d x = sin x + cos x 0 π 4 = 1 2 + 1 2 - 1 = 2 - 1 Sq. units

Sketch the curves and identify the region bounded by x = 1 2 , x = 2 , y = log e x and y = 2 x . Find the area of the region.

The given curves are x = 1 2 . . . ( i ) x = 2 . . . ( ii ) y = log e x . . . ( iii ) y = 2 x . . . ( iv ) Required area = Area of A B C D A = ∫ 1 / 2 2 2 x - log x d x = 2 x log 2 - ( x log x - x ) 1 / 2 2 = 4 log 2 - 2 log 2 + 2 - 2 log 2 - 1 2 log 1 2 + 1 2 = 4 - 2 log 2 - 5 2 log 2 + 3 2 sq . units .

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Gamma Question Bank for Engineering Mathematics>Class XII>Chapter 29 - Application of Integrals>Assignment Set - 1

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Gamma Question Bank for Engineering Mathematics>Chapter 29 - Application of Integrals>Assignment Set - 1>Q 7

$AOB$ is the positive quadrant of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ where $OA = a, OB = b$ . Then find the area between the arc $AB$ and the chord $AB$ of the ellipse.