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 problem requires finding a specific definite integral. The limits of integration, \( a \) and \( b \), are the coordinates of the intersection point of the parabola \( x^2 = 2y \) and the line \( y - 2x - 6 = 0 \) that lies in the second quadrant.

Concept Used:

The solution involves the following concepts:

1. Intersection of Curves: To find the intersection points of two curves, we solve their equations simultaneously.
2. Properties of Definite Integrals: For an integral with symmetric limits of integration, i.e., from \( -c \) to \( c \), the following property is very useful: \[ \int_{-c}^{c} f(x) \, dx = \int_{0}^{c} [f(x) + f(-x)] \, dx \]
3. Standard Integration: The integral of a power function is given by: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Step-by-Step Solution:

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

First, express \( y \) from the line equation:

\[ y = 2x + 6 \]

Substitute this expression for \( y \) into the equation of the parabola:

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

Rearrange this into a standard quadratic equation:

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

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

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

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

Now, find the corresponding \( y \) values:

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

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

The second quadrant is where \( x < 0 \) and \( y > 0 \). From the two points found, \( (-2, 2) \) is in the second quadrant.

Therefore, the point \( (a, b) \) is \( (-2, 2) \), which gives us the limits of integration: \( a = -2 \) and \( b = 2 \).

Step 3: Set up the integral with the determined 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 property of definite integrals for symmetric limits.

Let the integrand be \( f(x) = \frac{9x^2}{1 + 5^x} \). The limits are from -2 to 2, so we can use the property \( \int_{-c}^{c} f(x) \, dx = \int_{0}^{c} [f(x) + f(-x)] \, dx \).

First, find \( f(-x) \):

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

Now, find 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 Computation & Result

Now, we 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 value of the integral is 24.