Question

Let (a, b) be the point of intersection of the curve \(x^2 = 2y\) and the straight line \(y - 2x - 6 = 0\) in the second quadrant. Then the integral \(I = \int_{a}^{b} \frac{9x^2}{1+5^{x}} \, dx\) 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

The objective is to compute a specific definite integral. The integration bounds, denoted \( a \) and \( b \), correspond to the coordinates of the intersection point between the parabola defined by \( x^2 = 2y \) and the line \( y - 2x - 6 = 0 \) that is located in the second quadrant.

Key Concepts:

The solution employs the following principles:

1. Curve Intersection: To ascertain the intersection points of two curves, their respective equations are solved concurrently.
2. Definite Integral Properties: For integrals with symmetrical limits, i.e., ranging from \( -c \) to \( c \), the following property is highly beneficial: \[ \int_{-c}^{c} f(x) \, dx = \int_{0}^{c} [f(x) + f(-x)] \, dx \]
3. Basic Integration: The integral of a power function \( x^n \) is given by: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Solution Breakdown:

Step 1: Determine the intersection point of \( x^2 = 2y \) and \( y - 2x - 6 = 0 \).

From the linear equation, isolate \( y \):

\[ y = 2x + 6 \]

Substitute this into the parabolic equation:

\[ x^2 = 2(2x + 6) \] \[ x^2 = 4x + 12 \]

Rewrite as a standard quadratic equation:

\[ x^2 - 4x - 12 = 0 \]

Factor the quadratic to find \( x \) values:

\[ (x - 6)(x + 2) = 0 \]

The intersection occurs at \( x = 6 \) and \( x = -2 \).

Calculate the corresponding \( y \) values:

• For \( x = 6 \): \( y = 2(6) + 6 = 18 \). Point: \( (6, 18) \).
• For \( x = -2 \): \( y = 2(-2) + 6 = 2 \). Point: \( (-2, 2) \).

Step 2: Identify the intersection point \( (a, b) \) situated in the second quadrant.

The second quadrant is characterized by \( x < 0 \) and \( y > 0 \). Of the two points, \( (-2, 2) \) satisfies these conditions.

Therefore, \( (a, b) = (-2, 2) \), establishing the integration limits as \( a = -2 \) and \( b = 2 \).

Step 3: Formulate the integral using the established limits.

The integral to be evaluated is:

\[ I = \int_{a}^{b} \frac{9x^2}{1 + 5^x} \, dx = \int_{-2}^{2} \frac{9x^2}{1 + 5^x} \, dx \]

Step 4: Apply the definite integral property for symmetrical limits.

Let \( f(x) = \frac{9x^2}{1 + 5^x} \). With limits from -2 to 2, we apply \( \int_{-c}^{c} f(x) \, dx = \int_{0}^{c} [f(x) + f(-x)] \, dx \).

Determine \( f(-x) \):

\[ f(-x) = \frac{9(-x)^2}{1 + 5^{-x}} = \frac{9x^2}{1 + \frac{1}{5^x}} = \frac{9x^2}{\frac{5^x + 1}{5^x}} = \frac{9x^2 \cdot 5^x}{1 + 5^x} \]

Compute the sum \( f(x) + f(-x) \):

\[ f(x) + f(-x) = \frac{9x^2}{1 + 5^x} + \frac{9x^2 \cdot 5^x}{1 + 5^x} = \frac{9x^2(1 + 5^x)}{1 + 5^x} = 9x^2 \]

The integral simplifies to:

\[ I = \int_{0}^{2} (9x^2) \, dx \]

Final Calculation and Outcome

Evaluate the simplified integral:

\[ I = 9 \int_{0}^{2} x^2 \, dx \] \[ I = 9 \left[ \frac{x^3}{3} \right]_{0}^{2} \] \[ I = 3 [x^3]_{0}^{2} \] \[ I = 3 (2^3 - 0^3) = 3(8) = 24 \]

The integral evaluates to 24.