Question

The area of the region bounded by the curve \( y = 4x^3 - 6x^2 + 4x + 1 \) and the lines \( x = 1, x = 5 \) and the X axis is

• A. 428 sq. units
• B. 400 sq. units
• C. 334 sq. units
• D. 378 sq. units

Hint:

To find the area between a curve and the X-axis, set up the definite integral and solve for the area by evaluating the integral between the given limits.

Answer 1

The Correct Option is A

Step 1: Understanding the problem.
We are asked to find the area of the region bounded by the curve and the lines \( x = 1 \), \( x = 5 \), and the X-axis. The area can be found using the definite integral of the function between the limits \( x = 1 \) and \( x = 5 \).

Step 2: Setting up the integral.
The area is given by the integral: \[ \int_{1}^{5} \left( 4x^3 - 6x^2 + 4x + 1 \right) \, dx \] Solving this integral, we obtain the value \( 428 \, \text{sq. units} \).

Step 3: Conclusion.
Thus, the area of the region is 428 sq. units, which makes option (A) the correct answer.