Rainfall Volume Calculation?

📌 CONCEPT: The volume of water in the reservoir can be calculated using the definite integral of the area function, which represents the cross-sectional area of the water as a function of distance from the left end of the reservoir.

📐 RULE / FORMULA: The formula for the volume V is given by V = ∫[a,b] (length × width × depth) dx, where a and b are the limits of integration.

💡 WORKED EXAMPLE: To calculate the volume of water in the reservoir between x = 0 and x = 20 m, we first need to find the area function A(x) = (length × width × depth) = 100 × (5x + 2) × 3 = 300(5x + 2). Then, we integrate this function from x = 0 to x = 20 to get V = ∫[0,20] 300(5x + 2) dx = [150x^2 + 600x] from 0 to 20 = 150(20)^2 + 600(20) - 0 = 60000 m^3.

⚠️ COMMON MISTAKE: Students often forget to multiply the length, width, and depth when setting up the area function, or they incorrectly evaluate the definite integral by not applying the fundamental theorem of calculus.