The given integral is:

$\int_0^4 \int_{\frac{1}{4}x}^{\frac{3}{4}x} -22y e^x \, \mathrm{d}y \, \mathrm{d}x$

Step-by-Step Solution:

1. Integrate with respect to $y$: The inner integral involves $y$, so treat $e^x$ as constant during this step. The antiderivative of $-22y$ is:

   $\int -22y \, \mathrm{d}y = -11y^2$

   Applying the limits $y = \frac{1}{4}x$ to $y = \frac{3}{4}x$:

   $\int_{\frac{1}{4}x}^{\frac{3}{4}x} -22y \, e^x \, \mathrm{d}y = e^x \left[ -11 \left( \frac{3}{4}x \right)^2 + 11 \left( \frac{1}{4}x \right)^2 \right]$

   Simplify the expression inside the brackets:

   $= e^x \left[ -11 \frac{9}{16}x^2 + 11 \frac{1}{16}x^2 \right]$

   $= e^x \left[ -\frac{99}{16}x^2 + \frac{11}{16}x^2 \right]$

   $= e^x \left( -\frac{88}{16}x^2 \right)$

   $= -\frac{11}{2}x^2 e^x$

2. Integrate with respect to $x$: The outer integral becomes:

   $\int_0^4 -\frac{11}{2}x^2 e^x \, \mathrm{d}x$

   To solve this, use integration by parts. Let:

   $u = x^2 \quad \text{and} \quad \mathrm{d}v = e^x \, \mathrm{d}x$

   Then:

   $\mathrm{d}u = 2x \, \mathrm{d}x \quad \text{and} \quad v = e^x$

   Apply the integration by parts formula:

   $\int x^2 e^x \, \mathrm{d}x = x^2 e^x - \int 2x e^x \, \mathrm{d}x$

   For the remaining integral $\int 2x e^x \, \mathrm{d}x$, use integration by parts again. Let:

   $u = x \quad \text{and} \quad \mathrm{d}v = e^x \, \mathrm{d}x$

   Then:

   $\mathrm{d}u = \mathrm{d}x \quad \text{and} \quad v = e^x$

   Apply the formula:

   $\int x e^x \, \mathrm{d}x = x e^x - \int e^x \, \mathrm{d}x = x e^x - e^x$

   Substitute back:

   $\int x^2 e^x \, \mathrm{d}x = x^2 e^x - 2 \left( x e^x - e^x \right)$

   $= x^2 e^x - 2x e^x + 2e^x$

   Now substitute this into the outer integral:

   $\int_0^4 -\frac{11}{2}x^2 e^x \, \mathrm{d}x = -\frac{11}{2} \left[ \left( x^2 e^x - 2x e^x + 2e^x \right) \Big|_0^4 \right]$

   At $x = 4$:

   $4^2 e^4 - 2(4)e^4 + 2e^4 = 16e^4 - 8e^4 + 2e^4 = 10e^4$

   At $x = 0$:

   $0^2 e^0 - 2(0)e^0 + 2e^0 = 2$

   Therefore:

   $-\frac{11}{2} \left[ 10e^4 - 2 \right] = -\frac{11}{2} \cdot 10e^4 + \frac{11}{2} \cdot 2$

   $= -55e^4 + 11$

Final Answer:

$\boxed{-55e^4 + 11}$

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Would you like further details on any of the steps?

Here are some follow-up questions:

1. What is the purpose of integration by parts, and when is it used?
2. How do the bounds affect definite integrals?
3. Could a substitution simplify the outer integral?
4. Why do we separate constants like $e^x$ in the integration process?
5. What other methods can solve integrals involving products like $x^2 e^x$?

Tip: Always verify your integration steps by differentiating the result to ensure it matches the original integrand.