Questions: Find the area of the region between the curves 4x + y^2 = 12 and x = y.

Solution Steps

To find the area of the region between the curves \(4x + y^2 = 12\) and \(x = y\), we need to:

1. Express both equations in terms of \(y\) to find the points of intersection.
2. Determine the limits of integration by solving the equations simultaneously.
3. Set up the integral of the difference between the two functions over the interval found in step 2.
4. Evaluate the integral to find the area.

To find the area between the curves \(4x + y^2 = 12\) and \(x = y\), we first determine their points of intersection by solving the equations simultaneously. The intersection points are found to be \((-6, -6)\) and \((2, 2)\).

Next, we express \(x\) in terms of \(y\) for the curve \(4x + y^2 = 12\): \[ x = 3 - \frac{y^2}{4} \] The other curve is simply \(x = y\). The area \(A\) between the curves from \(y = -6\) to \(y = 2\) is given by the integral: \[ A = \int_{-6}^{2} \left( \left(3 - \frac{y^2}{4}\right) - y \right) \, dy \]

Calculating the integral: \[ A = \int_{-6}^{2} \left(3 - \frac{y^2}{4} - y\right) \, dy \] This evaluates to: \[ A = \frac{64}{3} \]

Final Answer