Question

The area bounded by the curve \(y = x^3\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is

• A. \(\dfrac{127}{4}\) sq. units
• B. \(64\) sq. units
• C. \(27\) sq. units
• D. \(\dfrac{255}{4}\) sq. units

Hint:

Area between a curve and the \(x\)-axis is obtained by definite integration of the function.

Answer 1

The Correct Option is D

Step 1: Write the expression for area.
Since the curve lies above the \(x\)-axis for \(x \ge 0\), the required area is \[ \int_{1}^{4} x^3 \, dx \]
Step 2: Integrate.
\[ \int x^3 dx = \frac{x^4}{4} \]
Step 3: Apply limits.
\[ \left[\frac{x^4}{4}\right]_{1}^{4} = \frac{256 - 1}{4} = \frac{255}{4} \]