To find the mass of a triangular region with vertices $(0,0)$, $(4,0)$, and $(0,2)$ and density function $\rho(x, y) = x^2 + y^2$, we need to set up and evaluate a double integral for the mass.

Step-by-Step Solution

1. Set Up the Region:
   The triangular region is bounded by:

   • The $x$-axis ($y = 0$),
   • The $y$-axis ($x = 0$),
   • The line from $(0,2)$ to $(4,0)$, which has the equation $y = -\frac{1}{2}x + 2$.

   Therefore, for any $x$ in $[0, 4]$, $y$ ranges from $0$ to $-\frac{1}{2}x + 2$.

2. Set Up the Integral for Mass: The mass $M$ of the region with density $\rho(x, y) = x^2 + y^2$ is given by: $M = \iint\limits_{\text{Region}} \rho(x, y) \, dA = \iint\limits_{\text{Region}} (x^2 + y^2) \, dA$

   Converting to an iterated integral, we get: $M = \int_0^4 \int_0^{-\frac{1}{2}x + 2} (x^2 + y^2) \, dy \, dx$

3. Evaluate the Inner Integral (with respect to $y$): $\int_0^{-\frac{1}{2}x + 2} (x^2 + y^2) \, dy = \left[ x^2 y + \frac{y^3}{3} \right]_0^{-\frac{1}{2}x + 2}$ Substituting $y = -\frac{1}{2}x + 2$: $= x^2 \left(-\frac{1}{2}x + 2\right) + \frac{\left(-\frac{1}{2}x + 2\right)^3}{3}$ Simplifying this expression yields a function of $x$, which we can then integrate with respect to $x$.

4. Evaluate the Outer Integral (with respect to $x$).

Since the solution provided in the image is approximately $13.333$, this value seems to be the final result after evaluating both integrals.

Would you like the details on evaluating each step, or any clarifications on specific parts of the process?

Related Questions

1. How would the mass change if the density function was $\rho(x, y) = x + y$ instead?
2. What would be the centroid of this triangular region if the density function remained $x^2 + y^2$?
3. How would the limits of integration change if the triangle’s vertices were different, say $(0, 0)$, $(3, 0)$, and $(0, 3)$?
4. How does the choice of density function affect the center of mass?
5. How can we find the moment of inertia for this region?

Tip:

When setting up double integrals, carefully identify the boundaries of the region and rewrite them in terms of inequalities for $x$ and $y$. This makes it easier to set up correct limits of integration.