soleve double integral from 0 to 1 and 0 to x^2 (x^2ydxdy)

To solve the double integral:

\[
\int_0^1 \int_0^{x^2} x^2 y \, dy \, dx
\]

We will first compute the inner integral with respect to \( y \), and then the outer integral with respect to \( x \).

### Step 1: Inner Integral

The inner integral is:

\[
\int_0^{x^2} x^2 y \, dy
\]

Since \( x^2 \) is a constant with respect to \( y \), we can factor it out:

\[
x^2 \int_0^{x^2} y \, dy
\]

Now, integrate \( y \) with respect to \( y \):

\[
\int y \, dy = \frac{y^2}{2}
\]

Now evaluate at the limits \( y = x^2 \) and \( y = 0 \):

\[
x^2 \left[ \frac{(x^2)^2}{2} - \frac{0^2}{2} \right] = x^2 \cdot \frac{x^4}{2} = \frac{x^6}{2}
\]

So the inner integral evaluates to:

\[
\frac{x^6}{2}
\]

### Step 2: Outer Integral

Now we compute the outer integral:

\[
\int_0^1 \frac{x^6}{2} \, dx
\]

We can factor out \( \frac{1}{2} \):

\[
\frac{1}{2} \int_0^1 x^6 \, dx
\]

The integral of \( x^6 \) is:

\[
\int x^6 \, dx = \frac{x^7}{7}
\]

Now evaluate at the limits \( x = 1 \) and \( x = 0 \):

\[
\frac{1}{2} \left[ \frac{1^7}{7} - \frac{0^7}{7} \right] = \frac{1}{2} \cdot \frac{1}{7} = \frac{1}{14}
\]

### Final Answer

The value of the double integral is:

\[
\frac{1}{14}
\]

Would you like further clarification on any step? Here are five related questions for practice:

1. How would you solve the integral if the limits of \( y \) were different?
2. What happens if the integrand was \( x^3 y \) instead of \( x^2 y \)?
3. How does changing the order of integration affect the result?
4. Can you solve a similar double integral where the outer limit is \( x^3 \) instead of \( x^2 \)?
5. How would you compute the integral if the limits of \( x \) were from 0 to 2?

**Tip:** Always check if you can simplify the integrand or bounds before integrating.

This detailed solution explains how to solve the double integral from 0 to 1 and 0 to x^2 for the function x^2y. The process involves computing inner and outer integrals, covering key calculus concepts like integration by parts and fundamental theorems of calculus. Suitable for university-level calculus students.

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