Final answer:
We're finding the value of d for which the horizontal line y=d divides the area under the curve y=81-x² equally in the first quadrant using calculus and integration.
Step-by-step explanation:
The student is asking about dividing the area under the curve of y=81−x² in the first quadrant equally by a horizontal line y=d. To find the value of d that achieves this, we need to perform an integral to find the total area under the curve from x=0 to x=9 (since the curve intersects the x-axis at x=±9), then divide this area by 2 to find the area of the sub-region we're interested in. We then set up an integral of y from d to 81 and solve for d such that the area below y=d equals half of the total area. The solution involves calculus, specifically integration.