Question

Area bounded by the curve y² (2a – x) = x³ and the line x = 2a is –

• A. 3pa²
• B. $\frac { 3 \pi \mathrm { a } ^ { 2 } } { 2 }$
• C. $\frac { 3 \pi \mathrm { a } ^ { 2 } } { 4 }$
• D. None of these

Answer 1

Answer: (A)

3pa²

Answer 2

y² (2a – x) = x³ is symmetrical about x-axis because even power of y is present. It also passes through (0, 0). The line x = 2a is asymptote as y ®  when x = 2a.

\ Area = 2 $\int _ { 0 } ^ { 2 \mathrm { a } } \mathrm { ydx }$ = 2 $\int _ { 0 } ^ { 2 a } \sqrt { \frac { x ^ { 3 } } { 2 a - x } }$ dx Put x = 2a sin² q

Ž dx = 4a sin q cos q dq

\ Area = 2$\int _ { 0 } ^ { \pi / 2 }$2a $\frac { \sin ^ { 3 } \theta } { \cos \theta }$ . 4a sin q cos q dq

= 16 a² $\int _ { 0 } ^ { \pi / 2 } \sin ^ { 4 }$q dq = 16 a² . $\frac { 3.1 } { 4.2 }$ $\frac { \pi } { 2 }$ = 3p a².