Question

Find the area bounded by the curves x = a cost, y = b sin t in the first quadrant

• A. $\frac { \pi a b } { 4 }$
• B. $\frac { \pi a ^ { 2 } b } { 4 }$
• C. $\frac { \pi a b ^ { 2 } } { 4 }$
• D. None of these

Answer 1

Answer: (A)

$\frac { \pi a b } { 4 }$

Answer 2

Clearly the given equation are the parametric equation of ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ . Curve meet the x-axis in the first quadrant at (a,0)

∴ Required area $= \int _ { \frac { \pi } { 2 } } ^ { 0 } ( b \sin t ) ( - a \cos t ) d t$

$= a b \int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } t d t = \left( \frac { \pi a b } { 4 } \right)$ $( \because$ At $x = 0 , t = \pi / 2$ and $x = a , t = 0 )$