Integrate[Integrate[\4xy-Power[y,3]\(41),{x,Power[x,3],Sqrt[x]}],{y,0,1}]

The expression you've provided is an iterated integral involving two variables, \(x\) and \(y\). The integral seems to include some notational issues. Let’s break it down and interpret the expression for you.

Your expression is:

\[
\int_0^1 \left( \int_{\sqrt{x}}^{x^3} 4xy - y^3 \, dx \right) dy
\]

Here’s how we can interpret this iterated integral:

1. **Inner integral**: \(\int_{\sqrt{x}}^{x^3} (4xy - y^3) \, dx\)
2. **Outer integral**: After evaluating the inner integral with respect to \(x\), we will integrate with respect to \(y\) over the interval \([0, 1]\).

### Step 1: Evaluate the inner integral
First, we integrate \(4xy - y^3\) with respect to \(x\).

\[
\int (4xy - y^3) \, dx = 2x^2y - y^3x
\]

Now, we evaluate this at the limits of integration \(\sqrt{x}\) and \(x^3\):

\[
\left[ 2x^2y - y^3x \right]_{\sqrt{x}}^{x^3}
\]

For the upper limit \(x^3\):

\[
2(x^3)^2y - y^3(x^3) = 2x^6y - y^3x^3
\]

For the lower limit \(\sqrt{x}\):

\[
2(\sqrt{x})^2y - y^3(\sqrt{x}) = 2xy - y^3\sqrt{x}
\]

Thus, the result of the inner integral is:

\[
\left( 2x^6y - y^3x^3 \right) - \left( 2xy - y^3\sqrt{x} \right)
\]
Simplifying:

\[
= 2x^6y - y^3x^3 - 2xy + y^3\sqrt{x}
\]

### Step 2: Outer integral
Now, we integrate the expression obtained from the inner integral with respect to \(y\), from \(0\) to \(1\):

\[
\int_0^1 \left( 2x^6y - y^3x^3 - 2xy + y^3\sqrt{x} \right) dy
\]

Each term can be integrated separately:

- For \(2x^6y\), we get:

\[
2x^6 \int_0^1 y \, dy = 2x^6 \cdot \frac{1}{2} = x^6
\]

- For \(-y^3x^3\), we get:

\[
-x^3 \int_0^1 y^3 \, dy = -x^3 \cdot \frac{1}{4} = -\frac{x^3}{4}
\]

- For \(-2xy\), we get:

\[
-2x \int_0^1 y \, dy = -2x \cdot \frac{1}{2} = -x
\]

- For \(y^3\sqrt{x}\), we get:

\[
\sqrt{x} \int_0^1 y^3 \, dy = \sqrt{x} \cdot \frac{1}{4} = \frac{\sqrt{x}}{4}
\]

### Step 3: Combine the results
Summing all the terms, the result of the outer integral is:

\[
x^6 - \frac{x^3}{4} - x + \frac{\sqrt{x}}{4}
\]

Thus, the final result of the iterated integral is:

\[
x^6 - \frac{x^3}{4} - x + \frac{\sqrt{x}}{4}
\]

Would you like any further clarification on the steps or concepts involved?

---

### Follow-up Questions:
1. How do we evaluate definite integrals for more complex functions?
2. What happens when we reverse the order of integration in double integrals?
3. How does the method of substitution affect double integrals?
4. Can we apply Fubini's theorem to simplify this integral?
5. How do iterated integrals change when integrating over polar coordinates?

**Tip**: When dealing with multiple integrals, breaking them down step-by-step and focusing on one variable at a time simplifies the process significantly.

Solve the double integral ∫₀¹(∫√xᵃ 4xy - y³ dx) dy using iterated integration. Explore step-by-step how to approach double integrals with variable limits, apply Fubini's Theorem, and compute definite integrals. Ideal for college students studying Multivariable Calculus.

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