Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the x-axis.

To find the area of the region bounded by the parabola, the tangent line, and the x-axis, we need to determine the points of intersection of these three curves.

Let's start by finding the point where the tangent line intersects the x-axis. Since the tangent line touches the parabola at (4, 64), the slope of the tangent line will be equal to the derivative of the parabola at x=4.

The derivative of y with respect to x can be found by differentiating y = 4x^2:
dy/dx = d/dx (4x^2) = 8x.

So, the derivative of y with respect to x is 8x. Evaluating the derivative at x=4:
dy/dx = 8(4) = 32.

Therefore, the slope of the tangent line to the parabola at (4, 64) is 32. Using the point-slope form of a linear equation, we have:
y - 64 = 32(x - 4).

Now, let's solve this equation for the point where the tangent line intersects the x-axis. When the y-coordinate is equal to zero (on the x-axis), we have:
0 - 64 = 32(x - 4).

Simplifying the equation further:
-64 = 32x - 128,
32x = 128 - 64,
32x = 64,
x = 2.

Therefore, the tangent line intersects the x-axis at x=2.

Now, we need to find the points of intersection of the parabola y = 4x^2 and the x-axis. To do this, we can set y equal to zero and solve for x:
4x^2 = 0,
x^2 = 0.

The only solution to this is x=0.

So, the parabola intersects the x-axis at x=0.

Now we have the three points of intersection:
A: (0, 0) - Intersection of the parabola and the x-axis.
B: (2, 0) - Intersection of the tangent line and the x-axis.
C: (4, 64) - Intersection of the parabola and the tangent line.

The region bounded by the parabola, the tangent line, and the x-axis is a triangle with base AC (the x-axis) and height BC (the line segment connecting points B and C).

To find the length of BC, we can use the distance formula:
BC = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Substituting the coordinates of B and C into the distance formula:
BC = sqrt((4 - 2)^2 + (64 - 0)^2)
= sqrt(2^2 + 64^2)
= sqrt(4 + 4096)
= sqrt(4100)
= 64.03124237432849.

The length of BC is approximately 64.0312.

Next, to find the area of the triangle, we can use the formula for the area of a triangle:
Area = 0.5 * base * height.

Substituting the values we have:
Area = 0.5 * 2 * 64.0312
= 64.0312.

Therefore, the area of the region bounded by the parabola y = 4x^2, the tangent line at (4, 64), and the x-axis is approximately 64.0312 units squared.