Fountain Height Equation?

📌 CONCEPT: The total height of the fountain can be found by integrating the height equation h(x) = 10x^3 - 12x^2 + 5x + 2 with respect to x, within the given limits of integration, which represent the horizontal distance from the fountain where the water jets merge.

📐 RULE / FORMULA: To find the total height, we use the definite integral ∫[a,b] h(x) dx, where a and b are the limits of integration, which in this case are 0 and 10, representing the start and end points of the fountain.

💡 WORKED EXAMPLE: To find the total height of the fountain when the water jets merge at a point 10 meters away, we evaluate the definite integral ∫[0,10] (10x^3 - 12x^2 + 5x + 2) dx. Using the power rule of integration, we get [10(1/4)x^4 - 12(1/3)x^3 + 5(1/2)x^2 + 2x] from 0 to 10.

⚠️ COMMON MISTAKE: Students often forget to apply the limits of integration, which can lead to incorrect results. Additionally, they may not correctly apply the power rule of integration, resulting in an incorrect antiderivative.