Question

Find the area enclosed by the curve \( y = x^3 - 19x + 30 \) and the X-axis.

• A. \( \frac{167}{2} \)
• B. \( \frac{517}{2} \)
• C. \( 36 \)
• D. \( 72 \)

Hint:

For finding enclosed areas, always determine the points of intersection and split the integral accordingly.

Answer 1

The Correct Option is B

Step 1: Finding the points where the curve intersects the X-axis. The given function is: \[ y = x^3 - 19x + 30 \] To find the x-intercepts, solve: \[ x^3 - 19x + 30 = 0 \] Using trial values and factorization, we get: \[ (x-3)(x-5)(x+2) = 0 \] Thus, the roots are: \[ x = -2, x = 3, x = 5 \] Step 2: Computing the enclosed area. The required area is given by: \[ A = \int_{-2}^{3} |x^3 - 19x + 30| dx + \int_{3}^{5} |x^3 - 19x + 30| dx \] Since the function changes sign at \( x = 3 \), we split the integral accordingly. Step 3: Evaluating the integral. Upon solving, the total enclosed area is: \[ A = \frac{517}{2} \]