1. Sketch the region:

First, sketch the curves y² = 2x and y = x - 4. This helps visualize the region whose area we need to find.

2. Find the points of intersection:

To find where the curves intersect, solve the equations simultaneously:

Substitute x = y + 4 into y² = 2x:

y² = 2(y + 4)

y² = 2y + 8

y² - 2y - 8 = 0

(y - 4)(y + 2) = 0

So, y = 4 or y = -2.

When y = 4, x = y + 4 = 4 + 4 = 8. Point of intersection: (8, 4)

When y = -2, x = y + 4 = -2 + 4 = 2. Point of intersection: (2, -2)

3. Set up the integral:

We will integrate with respect to y. The area A is given by:

A = ∫[from -2 to 4] (x_right - x_left) dy

Here, x_right is the x-value of the parabola y² = 2x, so x_right = y²/2.

And x_left is the x-value of the line y = x - 4, so x_left = y + 4.

Therefore, A = ∫[from -2 to 4] (y²/2 - (y + 4)) dy

4. Evaluate the integral:

A = ∫[from -2 to 4] (y²/2 - y - 4) dy

A = [y³/6 - y²/2 - 4y] evaluated from -2 to 4

A = [(4³/6 - 4²/2 - 4*4) - ((-2)³/6 - (-2)²/2 - 4*(-2))]

A = [(64/6 - 16/2 - 16) - (-8/6 - 4/2 + 8)]

A = [(32/3 - 8 - 16) - (-4/3 - 2 + 8)]

A = [32/3 - 24] - [-4/3 + 6]

A = 32/3 - 24 + 4/3 - 6

A = 36/3 - 30

A = 12 - 30

A = -18

Since area cannot be negative, we take the absolute value.