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:
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:
Show Hint
- 24
- 27
- 18
- 21
The Correct Option is A
Solution and Explanation
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:
- Intersection of Curves: To find the intersection points of two curves, we solve their equations simultaneously.
- Properties of Definite Integrals: For an integral with symmetric limits of integration, i.e., from \( -c \) to \( c \), the following property is very useful: \[ \int_{-c}^{c} f(x) \, dx = \int_{0}^{c} [f(x) + f(-x)] \, dx \]
- 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.
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