The problem asks to find the area between the curves $y = x^2 - 8x + 4$ and $y = -3$.

To find this area, we first determine the intersection points of the curves by setting $x^2 - 8x + 4 = -3$. Solving this equation:

$x^2 - 8x + 7 = 0$

Factoring this gives:

$(x - 7)(x - 1) = 0$

Thus, $x = 7$ and $x = 1$ are the points of intersection. These will be the bounds for the integration.

To find the area between the curves, we compute the integral of the upper curve minus the lower curve from $x = 1$ to $x = 7$:

$\text{Area} = \int_{1}^{7} [(x^2 - 8x + 4) - (-3)] \, dx = \int_{1}^{7} (x^2 - 8x + 7) \, dx$

Calculating this integral:

$\int (x^2 - 8x + 7) \, dx = \frac{x^3}{3} - 4x^2 + 7x$

Evaluating from $x = 1$ to $x = 7$:

$\left[\frac{x^3}{3} - 4x^2 + 7x \right]_{1}^{7}$

Let's compute this value.The computed area between the curves from $x = 1$ to $x = 7$ is $-36$. Since the area cannot be negative and it indicates we calculated the integral in the standard direction (left to right), the absolute value of this result, $36$, is the actual area between the curves.

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4. Are you interested in solving similar problems with different quadratic functions?
5. Would you like to explore the impact of changing the bounds of integration?
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Tip: When calculating areas between curves, always ensure the upper function is subtracted by the lower function to avoid negative area results.