https://file.aiquickdraw.com/mathbot/file/17283839168926o6cxvw4.jpg

Let's evaluate the given iterated integrals from the image.

### Problem (a):
\[
\int_0^3 \int_1^2 x^2 y \, dy \, dx
\]

### Step-by-step solution for (a):
1. First, solve the inner integral with respect to \( y \) (treating \( x^2 \) as a constant):
   \[
   \int_1^2 x^2 y \, dy = x^2 \int_1^2 y \, dy
   \]
   Now compute the integral of \( y \):
   \[
   \int_1^2 y \, dy = \left[\frac{y^2}{2}\right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}
   \]
   Therefore, the inner integral becomes:
   \[
   x^2 \cdot \frac{3}{2} = \frac{3x^2}{2}
   \]

2. Now, solve the outer integral with respect to \( x \):
   \[
   \int_0^3 \frac{3x^2}{2} \, dx = \frac{3}{2} \int_0^3 x^2 \, dx
   \]
   Compute the integral of \( x^2 \):
   \[
   \int_0^3 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^3 = \frac{3^3}{3} - \frac{0^3}{3} = 9
   \]
   Therefore, the outer integral becomes:
   \[
   \frac{3}{2} \cdot 9 = \frac{27}{2}
   \]
Thus, the value of the integral (a) is:
\[
\frac{27}{2}
\]

---

### Problem (b):
\[
\int_1^2 \int_0^3 x^2 y \, dx \, dy
\]

### Step-by-step solution for (b):
1. First, solve the inner integral with respect to \( x \) (treating \( y \) as a constant):
   \[
   \int_0^3 x^2 y \, dx = y \int_0^3 x^2 \, dx
   \]
   We already know from part (a) that:
   \[
   \int_0^3 x^2 \, dx = 9
   \]
   So, the inner integral becomes:
   \[
   y \cdot 9 = 9y
   \]

2. Now, solve the outer integral with respect to \( y \):
   \[
   \int_1^2 9y \, dy = 9 \int_1^2 y \, dy
   \]
   Compute the integral of \( y \):
   \[
   \int_1^2 y \, dy = \left[\frac{y^2}{2}\right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}
   \]
   Therefore, the outer integral becomes:
   \[
   9 \cdot \frac{3}{2} = \frac{27}{2}
   \]
Thus, the value of the integral (b) is also:
\[
\frac{27}{2}
\]

---

Let me know if you would like further details or have any questions!

Here are some related questions to consider:
1. How do you know when to switch the order of integration in iterated integrals?
2. How would you evaluate an iterated integral if the limits were dependent on each other?
3. Can you apply these techniques to triple integrals? How?
4. What changes if you have a function like \( x^2 + y^2 \) instead of \( x^2y \)?
5. How do the limits of integration affect the final answer?

**Tip:** When solving double integrals, always solve the inner integral first, and ensure that you treat the outer variable as a constant while solving the inner integral.

Evaluate the iterated integrals: (a) ∫ from 0 to 3 ∫ from 1 to 2 x^2 y dy dx, (b) ∫ from 1 to 2 ∫ from 0 to 3 x^2 y dx dy

This solution explains how to evaluate two iterated integrals: ∫ from 0 to 3 ∫ from 1 to 2 x^2 y dy dx and ∫ from 1 to 2 ∫ from 0 to 3 x^2 y dx dy, using step-by-step calculus methods. The solution covers key calculus concepts and is suitable for college-level students.

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