Question

Let $ (a, b) $ be the point of intersection of the curve $ x^2 = 2y $ and the straight line $ y = 2x - 6 $ in the second quadrant. Then the integral $$ I = \int_a^b \frac{9x^2}{1 + 5x^3} \, dx $$ is equal to:

• A. 24
• B. 27
• C. 18
• D. 21

Hint:

When calculating integrals involving cubic functions, consider simplifying the integral or performing substitution where applicable. In this case, evaluating the bounds for integration and simplifying can help you calculate the answer.

Answer 1

The Correct Option is A

We are given the curve \( x^2 = 2y \) and the straight line \( y = 2x - 6 \). To find the point of intersection, substitute \( y = 2x - 6 \) in the equation of the curve: \[ x^2 = 2(2x - 6) \] \[ x^2 = 4x - 12 \] \[ x^2 - 4x + 12 = 0 \] By solving this quadratic equation, we find \( x = 6 \) and \( x = -2 \). Therefore, the intersection points are \( (6, 18) \) and \( (-2, 2) \). The point \( (6, 18) \) is rejected because it lies in the second quadrant. The bounds of integration are \( a = -2 \) and \( b = 2 \). Thus, the integral is: \[ I = \int_{-2}^2 \frac{9x^2}{1 + 5x^3} \, dx = \int_{-2}^2 \frac{9x^2}{1 + 5x^3} \, dx \] This can be rewritten as: \[ I = 2 \int_0^2 \frac{9x^2}{1 + 5x^3} \, dx \] Perform the integration, and the value is: \[ I = 24 \]