Mass of a Thin Disk: A Simplified Approach?

📌 CONCEPT: The mass of a thin disk can be calculated using the formula for the surface area and density of the disk, which is a function of the radius and the density function.

📐 RULE / FORMULA: The mass of the disk is given by the integral of the density function multiplied by the area of the disk, which is πr^2. The formula is ∫[0,5] √x * π(5)^2 dx.

💡 WORKED EXAMPLE: To find the mass of the disk, we substitute the density function √x into the mass formula and evaluate the integral. The integral becomes ∫[0,5] √x * 25π dx. Using the power rule for integration, we get 25π * (2/3)x^(3/2) evaluated from 0 to 5. This simplifies to (25π/3) * 125.

⚠️ COMMON MISTAKE: Students often forget to evaluate the integral at the specified limits of integration or incorrectly apply the power rule for integration, leading to incorrect answers.