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We are tasked with finding the length of the curve formed by the intersection of two surfaces, defined by the following equations:

1. \( 4x^2 + y^2 = 36 \)
2. \( z = 6 - (x - 1)^2 - y^2 \)

### Step 1: Parametrize the curve

The first equation describes an ellipse, which can be parametrized using trigonometric functions. We can set:
\[
x = 3 \cos(t), \quad y = 6 \sin(t)
\]
where \( t \) is a parameter ranging from \( 0 \) to \( 2\pi \), covering the entire ellipse.

### Step 2: Substitute into the second equation

Now, substitute \( x = 3 \cos(t) \) and \( y = 6 \sin(t) \) into the second equation to express \( z \) as a function of \( t \):
\[
z = 6 - (3\cos(t) - 1)^2 - (6\sin(t))^2
\]

### Step 3: Compute the length of the curve

The formula for the length of a space curve \( (x(t), y(t), z(t)) \) is:
\[
L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2} \, dt
\]

Now we compute the derivatives \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \):
\[
\frac{dx}{dt} = -3 \sin(t)
\]
\[
\frac{dy}{dt} = 6 \cos(t)
\]
Next, we differentiate \( z(t) \) with respect to \( t \).

Let's compute that explicitly to find the length of the curve.The symbolic calculation of the integral took too long. Instead, I'll approximate the length numerically using the parametrization and derivatives computed earlier.

Let me proceed with that method.The length of the curve formed by the intersection of the given surfaces is approximately **115.3 units**, rounded to one decimal place.

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:
1. How is the parametric equation for an ellipse derived?
2. What are other methods to compute the length of a space curve?
3. Can the length of a curve always be computed exactly?
4. How does the choice of parametrization affect curve length calculations?
5. What is the role of the gradient in surface intersection problems?

**Tip**: For complex surface intersections, converting to parametric equations often simplifies the curve length calculation.

Hallar la longitud de la curva de intersección de las superficies determinadas por las ecuaciones 4x^2 + y^2 = 36 y z = 6 - (x - 1)^2 - y^2

Solve the problem of finding the length of the curve formed by the intersection of the ellipse defined by 4x^2 + y^2 = 36 and the surface z = 6 - (x - 1)^2 - y^2. This solution involves parametric equations and space curve length calculations. Suitable for advanced calculus students.

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