The problem you provided is asking to compute a double integral:

$\int \int_D (9x + 4y) \, dA$

Where the region $D$ is bounded by the curves $y = 6x - x^2$ and $y = x^2$.

Let me break this down into steps to help you solve it.

Steps to solve:

1. Understand the Region $D$:

   • You need to find the points where the curves $y = 6x - x^2$ and $y = x^2$ intersect. This will define the limits of integration for the double integral.
   • Set $6x - x^2 = x^2$, and solve for $x$.

2. Set Up the Integral:

   • The outer integral will be over the $x$-limits from the intersection points.
   • The inner integral will be over the $y$-limits, between the curve $y = x^2$ (lower boundary) and $y = 6x - x^2$ (upper boundary).

3. Evaluate the Integral:

   • Perform the integration step by step.

Let me proceed with these steps and show you the solution. I'll start by calculating the points of intersection between the two curves. sympy import symbols, Eq, solve

Define variables

x = symbols('x')

Define the two curves

curve1 = 6*x - x2 # y = 6x - x^2 curve2 = x2 # y = x^2

Set the two curves equal to find intersection points

equation = Eq(curve1, curve2)

Solve for x

intersection_points = solve(equation, x) intersection_pointsThe two curves $y = 6x - x^2$ and $y = x^2$ intersect at $x = 0$ and $x = 3$.

Thus, the region $D$ is bounded by $x = 0$ and $x = 3$.

Next Steps:

1. Set up the integral: The double integral will have limits for $x$ from 0 to 3. The inner integral will have $y$ ranging from $y = x^2$ to $y = 6x - x^2$.

The integral can be written as:

$\int_0^3 \left( \int_{x^2}^{6x - x^2} (9x + 4y) \, dy \right) dx$

Let me now compute the integral step by step. sympy import integrate

Define the integrand (9x + 4y)

y = symbols('y') integrand = 9x + 4y

Inner integral with respect to y, from y = x^2 to y = 6x - x^2

inner_integral = integrate(integrand, (y, x2, 6*x - x2))

Outer integral with respect to x, from x = 0 to x = 3

double_integral = integrate(inner_integral, (x, 0, 3)) double_integral.simplify()The value of the double integral is:

$\int \int_D (9x + 4y) \, dA = \frac{567}{2}$

This is the result for the given region $D$.

If you have any questions or need further details, feel free to ask!

Additional questions you might explore:

1. How do you visualize the region $D$ defined by these curves?
2. What happens if you change the integrand to a different expression, say $9x + y^2$?
3. How would the result change if the curves were $y = x^3$ and $y = 6x - x^2$?
4. What are the geometric interpretations of double integrals over bounded regions?
5. How would you evaluate a similar double integral with polar coordinates?

Tip:

When working with double integrals, it's crucial to carefully determine the limits of integration based on the region $D$ you're integrating over. The shape and bounds of the region often guide the integration process.