Consider the following iterated integral $$ \int_{R} \int f(x, y) d A $$. $$ \int_{0}^{30} \int_{y / 2}^{15}(x+y) d x d y $$

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1. Write the given iterated integral:
   $$
   \int_{0}^{30} \int_{y/2}^{15} (x+y) \, dx \, dy
   $$

2. Evaluate the inner integral with respect to $$x$$:
   $$
   \int_{y/2}^{15} (x+y) \, dx
   $$
   Split the integral into two parts:
   $$
   \int_{y/2}^{15} x \, dx + \int_{y/2}^{15} y \, dx
   $$
   
   For the first term:
   $$
   \int_{y/2}^{15} x \, dx = \left[ \frac{x^2}{2} \right]_{x=y/2}^{15} = \frac{15^2}{2} - \frac{(y/2)^2}{2} = \frac{225}{2} - \frac{y^2}{8}
   $$
   
   For the second term (since $$y$$ is constant with respect to $$x$$):
   $$
   \int_{y/2}^{15} y \, dx = y \left[ x \right]_{y/2}^{15} = y\left(15 - \frac{y}{2}\right) = 15y - \frac{y^2}{2}
   $$
   
   Combining both results:
   $$
   \int_{y/2}^{15} (x+y) \, dx = \frac{225}{2} - \frac{y^2}{8} + 15y - \frac{y^2}{2}
   $$
   
   Combine like terms for $$y^2$$ by writing $$\frac{y^2}{2}$$ as $$\frac{4y^2}{8}$$:
   $$
   -\frac{y^2}{8} - \frac{4y^2}{8} = -\frac{5y^2}{8}
   $$
   Hence, the inner integral becomes:
   $$
   \frac{225}{2} + 15y - \frac{5y^2}{8}
   $$

3. Substitute the result into the outer integral:
   $$
   \int_{0}^{30} \left( \frac{225}{2} + 15y - \frac{5y^2}{8} \right) dy
   $$
   
4. Integrate each term with respect to $$y$$:

- For the constant term:
     $$
     \int_{0}^{30} \frac{225}{2} \, dy = \frac{225}{2} y \Big|_{0}^{30} = \frac{225}{2} \times 30 = 225 \times 15 = 3375
     $$
     
   - For the linear term:
     $$
     \int_{0}^{30} 15y \, dy = \frac{15y^2}{2} \Big|_{0}^{30} = \frac{15 \times (30)^2}{2} = \frac{15 \times 900}{2} = \frac{13500}{2} = 6750
     $$
     
   - For the quadratic term:
     $$
     \int_{0}^{30} \frac{-5y^2}{8} \, dy = -\frac{5}{8} \int_{0}^{30} y^2 \, dy = -\frac{5}{8} \cdot \frac{y^3}{3} \Big|_{0}^{30} = -\frac{5}{24}(30)^3
     $$
     Since $$(30)^3 = 27000$$,
     $$
     -\frac{5}{24} \times 27000 = -\frac{135000}{24} = -5625
     $$

5. Combine the results of the three integrals:
   $$
   3375 + 6750 - 5625 = 4500
   $$

The value of the iterated integral is:
$$
4500
$$
The value of the iterated integral is 4500.

Consider the following iterated integral .

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