Let the area of the region enclosed by the curves y=3 x, 2 y=27-3 x and y=3 x-x x be A . Then 10 A is equal to

Correct answer: B. 162 y=3 x, 2 y=27-3 x \& y=3 x-x x aligned & A=∫_0^3 3 x-(3 x-x x ) dx+∫_3^9( 27-3 x 2 -(3 x-x x )) dx & A=∫_0^3 x^ 3 / 2 dx+∫_3^9 (27)/(2)- 9 x 2 +x^ 3 / 2 dx & A=[ 2 x^ 5 / 2 5 ]_0^3+(27)/(2)[x]_3^9-(9)/(2)[ x^2 2 ]_3^9+[ 2 x^ 5 / 2 5 ]_3^9 & A=(2)/(5)(3^ 5 / 2 )+(27)/(2)(6)-(9)/(4)(72)+(2)/(5)(9^ 5 / 2 -3^ 5 / 2 ) & A=(2)/(5)(3^ 5 / 2 )+81-162+(2)/(5) × 3^5-(2)/(5) × 3^ 5 / 2 & A=(486)/(5)-81=(81)/(5) & 10 A=162 & Ans. =(2) aligned

JEE Main 20246 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Area Under Curves

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Area Under Curves

Let the area of the region enclosed by the curves y=3 x, 2 y=27-3 x...

Let the area of the region enclosed by the curves

y=3 x, 2 y=27-3 x

and

y=3 x-x \sqrt{x}

A

. Then

10 A

is equal to

Options

A172
B162
C154
D184

Correct answer

B. 162

Step-by-step solution

y=3 x, 2 y=27-3 x \& y=3 x-x \sqrt{x}

\begin{aligned} & \mathrm{A}=\int_0^3 3 \mathrm{x}-(3 \mathrm{x}-\mathrm{x} \sqrt{\mathrm{x}}) \mathrm{dx}+\int_3^9\left(\frac{27-3 \mathrm{x}}{2}-(3 \mathrm{x}-\mathrm{x} \sqrt{\mathrm{x}})\right) \mathrm{dx} \\ & \mathrm{A}=\int_0^3 \mathrm{x}^{3 / 2} \mathrm{dx}+\int_3^9 \frac{27}{2}-\frac{9 \mathrm{x}}{2}+\mathrm{x}^{3 / 2} \mathrm{dx} \\ & \mathrm{A}=\left[\frac{2 \mathrm{x}^{5 / 2}}{5}\right]_0^3+\frac{27}{2}[\mathrm{x}]_3^9-\frac{9}{2}\left[\frac{\mathrm{x}^2}{2}\right]_3^9+\left[\frac{2 \mathrm{x}^{5 / 2}}{5}\right]_3^9 \\ & \mathrm{ A}=\frac{2}{5}\left(3^{5 / 2}\right)+\frac{27}{2}(6)-\frac{9}{4}(72)+\frac{2}{5}\left(9^{5 / 2}-3^{5 / 2}\right) \\ & \mathrm{A}=\frac{2}{5}\left(3^{5 / 2}\right)+81-162+\frac{2}{5} \times 3^5-\frac{2}{5} \times 3^{5 / 2} \\ & \mathrm{ A}=\frac{486}{5}-81=\frac{81}{5} \\ & 10 \mathrm{ A}=162 \\ & \text{Ans. }=(2)\end{aligned}

Practice Area Under Curves on Quantrex Academy →

More from Area Under Curves