Evaluating Total Revenue?

📌 CONCEPT: The total revenue can be found by integrating the revenue function R(x) over a given interval, representing the number of units sold.

📐 RULE / FORMULA: The total revenue is given by the definite integral of R(x) from a to b, i.e., ∫[a,b] R(x) dx, where a and b are the initial and final units sold.

💡 WORKED EXAMPLE: Consider the revenue function R(x) = 3x^2 + 5x - 4. To find the total revenue generated from the sale of the first 500 units, we evaluate ∫[0,500] (3x^2 + 5x - 4) dx. We can break this down into two parts: ∫[0,400] (3x^2 + 5x - 4) dx and ∫[400,500] (3x^2 + 5x - 4) dx, to validate our approach. After evaluating the integrals, we add the results to find the total revenue.

⚠️ COMMON MISTAKE: Students may forget to apply the Fundamental Theorem of Calculus to evaluate the definite integral, or mistakenly integrate the revenue function over the wrong interval.