Final answer:
To find the positive value of k for the function g(x) = k^2 - x^2 with area 500/3 under the curve, use the integral of g(x) from 0 to k. The calculated value of k is approximately 5.42.
Step-by-step explanation:
The question asks to find the positive value of k such that the area bounded by the graph of the function g(x) = k^2 - x^2 and the x-axis is 500/3. To solve this, we use the concept of definite integrals to find the area under the curve. The function represents a downward-opening parabola, which intersects the x-axis at points x = -k and x = k. The area under one half of the parabola (from 0 to k) is represented by the integral of g(x) from 0 to k, which gives us (1/2)k^3. Since we want the total area under the entire parabola to be 500/3, we need to consider the area under both halves (from -k to k), which is twice the integral from 0 to k: 2(1/2)k^3 = k^3. Setting this equal to 500/3 and solving for k yields:
k^3 = 500/3
k = ∛(500/3)
k ≈ 5.42 (to two decimal places)
Therefore, the positive value of k that gives the area of 500/3 under the function g(x) is approximately 5.42.