Area bounded by the curve \( y=x^3 \) and the line \( y=4x \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8 Square units
The curves meet where \( x^3=4x \), that is at x = −2, 0, 2. The region has two equal parts, symmetric about the origin. For 0 < x < 2, \( 4x\gt x^3 \).
\( \displaystyle \text{Area}=2\int_0^2(4x-x^3)\,dx \)
\( \displaystyle =2\left[2x^2-\frac{x^4}{4}\right]_0^2 \)
\( =2(8-4) \)
\( =8 \) square units
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{4} \) Square units — Incorrect. The area is 8, not \( \frac{1}{4} \).
Option (3) \( \displaystyle\frac{1}{8} \) Square units — Incorrect. The area is 8, not \( \frac{1}{8} \).
Option (4) 4 Square units — Incorrect. This is only one of the two equal parts.