Use the form of the definition of the integral to evaluate the intergral.
From 0 to 9 (10+6x-x^2)
To evaluate the integral of the function f(x) = 10 + 6x - x^2 from 0 to 9 using the definition of the integral, we can follow these steps:
Step 1: Rewrite the function f(x) = 10 + 6x - x^2 in the form of the definition of the integral, which is ∫f(x) dx.
Step 2: Expand the function and break it down into separate terms:
f(x) = 10 + 6x - x^2 = 10 + 6x + (-x^2)
Step 3: Integrate each term separately:
∫10 dx = 10x
∫6x dx = 3x^2
∫-x^2 dx = -1/3x^3
Step 4: Combine the integral of each term to evaluate the definite integral:
∫f(x) dx = ∫(10 + 6x - x^2) dx
= 10x + 3x^2 - 1/3x^3
Step 5: Evaluate the definite integral from 0 to 9:
∫[0 to 9](10 + 6x - x^2) dx = [10x + 3x^2 - 1/3x^3] [0 to 9]
= [10(9) + 3(9)^2 - 1/3(9)^3] - [10(0) + 3(0)^2 - 1/3(0)^3]
= [90 + 243 - 243/3] - [0 + 0 - 0]
= [90 + 243 - 81] - [0 + 0 - 0]
= 252
Therefore, the value of the definite integral of the function f(x) = 10 + 6x - x^2 from 0 to 9 is equal to 252.