Final answer:
To find the area under the graph of the function f(x) = (50/6)x³ between x = 2.5 and x = 7.5, we need to calculate the definite integral of the function from 2.5 to 7.5.
Step-by-step explanation:
First, let's graph the function f(x) = (50/6)x³. To find P(2.5 < x < 7.5), we need to find the area under the graph between x = 2.5 and x = 7.5. To do this, we can calculate the definite integral of the function from 2.5 to 7.5.
Using the definite integral formula, we have ∫(2.5 to 7.5) (50/6)x³ dx. This can be solved by finding the antiderivative of (50/6)x³, which is (50/24)x⁴. Evaluating this antiderivative from 2.5 to 7.5 gives us the area under the graph between x = 2.5 and x = 7.5.
P(2.5 < x < 7.5) = ∫(2.5 to 7.5) (50/6)x³ dx = [ (50/24)x⁴ ] from 2.5 to 7.5 = (50/24)(7.5⁴ - 2.5⁴).