Key Concepts and Formulas

1. Solving Simultaneous Equations: To find the point(s) of intersection of curves, we solve their equations concurrently.
2. Cartesian Quadrants: The second quadrant is defined by points with negative x-coordinates and positive y-coordinates.
3. Definite Integral Property (King's Rule): For a continuous function $f(x)$ on $[p, q]$, $\int_p^q f(x) \, dx = \int_p^q f(p+q-x) \, dx$. This is particularly useful for integrals with symmetric limits or integrands involving exponential/trigonometric functions.

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Step-by-Step Solution

Step 1: Find the Point of Intersection $(a,b)$

We are given the equations of a curve and a line:

1. $x^2 = 2y$
2. $y - 2x - 6 = 0$

We need to find the point of intersection $(a,b)$ that lies in the second quadrant.

• Explanation: To find the intersection, we substitute the expression for $y$ from the first equation into the second equation. From $x^2 = 2y$, we get $y = \frac{x^2}{2}$. Substituting this into the second equation: $\frac{x^2}{2} - 2x - 6 = 0$ Multiply by 2 to clear the fraction: $x^2 - 4x - 12 = 0$ Factor the quadratic equation: $(x-6)(x+2) = 0$ This gives two possible x-values: $x=6$ or $x=-2$.

  Now, find the corresponding y-values using $y = \frac{x^2}{2}$: If $x=6$, then $y = \frac{6^2}{2} = \frac{36}{2} = 18$. The point is $(6, 18)$. If $x=-2$, then $y = \frac{(-2)^2}{2} = \frac{4}{2} = 2$. The point is $(-2, 2)$.

• Identify the point in the second quadrant: The second quadrant has $x < 0$ and $y > 0$. The point $(-2, 2)$ satisfies these conditions. Therefore, $(a, b) = (-2, 2)$, which means $a = -2$ and $b = 2$.

Step 2: Set up the Definite Integral

The integral to be evaluated is $\mathrm{I}=\int_{\mathrm{a}}^{\mathrm{b}} \frac{9 x^2}{1+5^x} \mathrm{~d} x$. Substituting the values of $a$ and $b$: $\mathrm{I}=\int_{-2}^{2} \frac{9 x^2}{1+5^x} \mathrm{~d} x \quad \text{..... (1)}$

Step 3: Apply King's Rule to Evaluate the Integral

• Explanation: The integral has symmetric limits $[-2, 2]$. This suggests using King's Rule, which states $\int_p^q f(x) \, dx = \int_p^q f(p+q-x) \, dx$. Here, $p=-2$ and $q=2$, so $p+q = -2+2=0$. Let $f(x) = \frac{9x^2}{1+5^x}$. Applying King's Rule, we replace $x$ with $p+q-x = 0-x = -x$: $\mathrm{I}=\int_{-2}^{2} \frac{9 (-x)^2}{1+5^{-x}} \mathrm{~d} x$ Since $(-x)^2 = x^2$, we have: $\mathrm{I}=\int_{-2}^{2} \frac{9 x^2}{1+5^{-x}} \mathrm{~d} x \quad \text{..... (2)}$

  Now, simplify the denominator $1+5^{-x}$: $1+5^{-x} = 1 + \frac{1}{5^x} = \frac{5^x+1}{5^x}$ Substitute this back into equation (2): $\mathrm{I}=\int_{-2}^{2} \frac{9 x^2}{\frac{5^x+1}{5^x}} \mathrm{~d} x = \int_{-2}^{2} \frac{9 x^2 \cdot 5^x}{1+5^x} \mathrm{~d} x \quad \text{..... (3)}$

Step 4: Combine the Integral Expressions

• Explanation: Add equation (1) and equation (3). This is a standard technique when applying King's Rule for integrals of this form. $\mathrm{I} + \mathrm{I} = \int_{-2}^{2} \frac{9 x^2}{1+5^x} \mathrm{~d} x + \int_{-2}^{2} \frac{9 x^2 \cdot 5^x}{1+5^x} \mathrm{~d} x$ $2\mathrm{I} = \int_{-2}^{2} \left( \frac{9 x^2}{1+5^x} + \frac{9 x^2 \cdot 5^x}{1+5^x} \right) \mathrm{~d} x$ Combine the terms inside the integral since they have a common denominator: $2\mathrm{I} = \int_{-2}^{2} \frac{9 x^2 (1 + 5^x)}{1+5^x} \mathrm{~d} x$ The term $(1+5^x)$ cancels out: $2\mathrm{I} = \int_{-2}^{2} 9 x^2 \mathrm{~d} x$

Step 5: Evaluate the Simplified Integral

• Explanation: Now we have a simple polynomial integral. $2\mathrm{I} = 9 \int_{-2}^{2} x^2 \mathrm{~d} x$ The integral of $x^2$ is $\frac{x^3}{3}$. $2\mathrm{I} = 9 \left[ \frac{x^3}{3} \right]_{-2}^{2}$ Evaluate at the limits: $2\mathrm{I} = 9 \left( \frac{2^3}{3} - \frac{(-2)^3}{3} \right)$ $2\mathrm{I} = 9 \left( \frac{8}{3} - \frac{-8}{3} \right)$ $2\mathrm{I} = 9 \left( \frac{8}{3} + \frac{8}{3} \right)$ $2\mathrm{I} = 9 \left( \frac{16}{3} \right)$ $2\mathrm{I} = 3 \times 16$ $2\mathrm{I} = 48$ Solve for I: $\mathrm{I} = \frac{48}{2}$ $\mathrm{I} = 24$

Common Mistakes & Tips

• Quadrant Identification: Ensure you correctly identify the second quadrant ($x<0, y>0$) when determining the values of $a$ and $b$.
• King's Rule Application: Remember that King's Rule is most effective when the sum of the limits ($p+q$) is a simple constant, especially 0, which simplifies $p+q-x$ to $-x$.
• Algebraic Simplification: Be careful with algebraic manipulations, especially when dealing with fractions and exponential terms like $5^{-x}$.

Summary

The problem requires finding the intersection point of a parabola and a line, then evaluating a definite integral. We first solved the system of equations to find the intersection point $(-2, 2)$ in the second quadrant, establishing the limits of integration as $a=-2$ and $b=2$. The integral $\mathrm{I}=\int_{-2}^{2} \frac{9 x^2}{1+5^x} \mathrm{~d} x$ was then evaluated using King's Rule ($\int_p^q f(x) \, dx = \int_p^q f(p+q-x) \, dx$). By applying this rule with $p=-2$ and $q=2$, we obtained a second expression for I, and summing the original and transformed integrals simplified the integrand significantly. This led to the evaluation of a straightforward polynomial integral, yielding the final result.

The final answer is \boxed{24}.