Final answer:
The area under the curve of f(x)=5x⁵ between x=18 and x=20 is found by calculating the definite integral from 18 to 20, using the power rule of integration and simplifying the result.
Step-by-step explanation:
To find the area under the curve of the function f(x)=5x⁵ between x=18 and x=20 using the fundamental theorem of calculus, we need to calculate the definite integral of f(x) from 18 to 20. This process will give us the total area under the function's curve within the specified range.
We can set up the integral as follows:
∫₁₈¹¹ 5x⁵ dx
To solve this integral, we use the power rule of integration, which states that ∫ x^n dx = ⅓(x^(n+1))/(n+1) for any real number n that is not equal to -1.
Therefore, the integral becomes:
5 ∫₁₈¹¹ x⁵ dx = (5/6) * [x⁶]¹¹₁₈ = (5/6) * (20⁶ - 18⁶)
After computing the powers and the subtraction, we multiply by 5/6 to find the final area under the curve.