There are two points P=(a,a^2) and Q=(b,b^2) (a<b) moving along the parabola y=x^2. If the area of the region bounded by y=x^2 and the line segment PQ is always 36, what is the value of lim as a→∞ of PQ/a?

To find the value of lim as a → ∞ of PQ/a, we need to determine the relationship between the length of the line segment PQ and the value of a.

Let's start by finding the equation of the line segment PQ. The coordinates of point P are (a, a^2), and the coordinates of point Q are (b, b^2). Since both points lie on the parabola y=x^2, we equate the y-values to the square of the x-values:

a^2 = (a^2)^2
b^2 = (b^2)^2

Simplifying these equations, we get:
a^2 = a^4
b^2 = b^4

Taking the square root of both sides, we find:
a = ± a^2
b = ± b^2

Since a < b, we consider only the positive roots. So, a = a^2 and b = b^2.

To find the intersection points a and b, we equate the equations:
a = a^2
b = b^2

Simplifying further, we have:
a^2 - a = 0
b^2 - b = 0

Factoring out a and b, we get:
a(a - 1) = 0
b(b - 1) = 0

From these equations, we have two possibilities:
1) Either a = 0 or a - 1 = 0
2) Either b = 0 or b - 1 = 0

Since a < b, the only valid solution is a - 1 = 0, which gives us a = 1.

Now that we have found the intersection point between the line segment PQ and the parabola y=x^2, we can proceed to calculate the length of line segment PQ.

The length of a line segment can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the coordinates of points P and Q, we get:
d = sqrt((b - a)^2 + (b^2 - a^2)^2)

Since we know that a = 1, the equation simplifies to:
d = sqrt((b - 1)^2 + (b^2 - 1)^2)

Now, we need to find the value of b that satisfies the condition where the area of the region bounded by y=x^2 and the line segment PQ is always 36.

The area of the region bounded by y=x^2 and the line segment PQ can be calculated using integration:
A = ∫[a,b] (x^2) dx - ∫[a,b] (line segment PQ) dx

Since the area is constant and given as 36, we have the equation:
36 = ∫[a,b] (x^2) dx - ∫[a,b] (line segment PQ) dx

To solve for b, we need to integrate the expression for (line segment PQ) dx.

Substituting the equation of the line segment PQ as y = x^2, we can rewrite it as:
(line segment PQ) dx = (x^2) dx

Thus, our equation becomes:
36 = ∫[a,b] (x^2) dx - ∫[a,b] (x^2) dx

Simplifying further, we get:
36 = 0

This equation is not valid because 36 ≠ 0. Hence, there is no value of b that satisfies the given condition.

Therefore, there is no well-defined value for the limit as a → ∞ of PQ/a.