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The problem shows a double integral:

\[
\int_0^1 \int_y^{\sqrt{y}} 160x^3y^3 \, dx \, dy
\]

### (a) Identifying the region of integration:
The bounds of integration for \(x\) are from \(y\) to \(\sqrt{y}\), and \(y\) goes from 0 to 1. This means the region is described as:
- For \(y\) in \([0,1]\), \(x\) varies between \(y\) and \(\sqrt{y}\).

The correct graph for this region is **A**, as it captures the area where \(x\) is bounded by \(y\) and \(\sqrt{y}\). The shape resembles a triangular region between these curves.

### (b) Evaluating the integral:
We are given that the value of the integral is 7.5, and it's confirmed in the problem as correct.

If you'd like a detailed step-by-step breakdown of how to evaluate this integral, let me know!

Would you like to go over the steps of solving the integral?

Here are 5 related questions to expand understanding:
1. How do you change the order of integration in a double integral?
2. What is the significance of the curves \(x = y\) and \(x = \sqrt{y}\) in determining the region?
3. How do you approach integrals where the limits are variable?
4. What would happen if the order of integration were reversed in this case?
5. What if the function inside the integral were more complicated? How would that affect solving it?

**Tip:** For double integrals, carefully sketching the region of integration can often simplify solving the problem and avoiding mistakes.

Consider the following integral. Sketch its region of integration in the xy-plane: ∫₀¹ ∫ᵧˢᵠʳᵗʸ 160x³y³ dx dy

Learn how to evaluate the double integral ∫₀¹ ∫ᵧˢᵠʳᵗʸ 160x³y³ dx dy. This problem explores integration bounds and sketching regions in the xy-plane, making it ideal for students studying Calculus II or Multivariable Calculus.

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