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The given problem asks to evaluate the double integral:

\[
\iint_{R} (4y - 1) \, dA
\]

where \( R = [0, 5] \times [0, 4] \), meaning that \( x \) ranges from 0 to 5 and \( y \) ranges from 0 to 4.

We can solve this step-by-step.

### Step 1: Set up the integral
The integral can be written as:

\[
\int_0^5 \int_0^4 (4y - 1) \, dy \, dx
\]

### Step 2: Evaluate the inner integral
We first integrate with respect to \( y \):

\[
\int_0^4 (4y - 1) \, dy = \left[ 2y^2 - y \right]_0^4
\]

Now, evaluate this at the bounds \( y = 4 \) and \( y = 0 \):

\[
\left( 2(4)^2 - (4) \right) - \left( 2(0)^2 - (0) \right) = (2 \cdot 16 - 4) = 32 - 4 = 28
\]

So, the result of the inner integral is 28.

### Step 3: Evaluate the outer integral
Now we integrate with respect to \( x \):

\[
\int_0^5 28 \, dx = 28x \big|_0^5 = 28(5) - 28(0) = 140
\]

### Final Answer:
The value of the double integral is:

\[
\boxed{140}
\]

Would you like further details or have any questions?

### Related Questions:
1. What is the geometrical interpretation of a double integral?
2. How would the answer change if the function was different, such as \( (2y - 3) \)?
3. Can you evaluate the same integral over a different region, such as \( R = [1, 4] \times [0, 3] \)?
4. How does changing the limits of integration affect the solution?
5. Can you perform the integration using polar coordinates for a circular region?

### Tip:
When solving double integrals, always make sure to carefully choose the correct order of integration and verify the bounds for each variable.

Evaluate \(\iint_{R} (4y - 1) \, dA\) where \(R = [0, 5] \times [0, 4]\)

This example evaluates the double integral of the function (4y - 1) over the rectangular region defined by R = [0, 5] x [0, 4]. Follow a step-by-step solution involving multivariable calculus and Fubini's Theorem, suitable for college-level calculus students.

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