Chapter 6 | Applications of Integration
687
Figure 6.50 (a) A thin disk in the xy -plane. (b) A representative washer.
We now approximate the density and area of the washer to calculate an approximate mass, m i . Note that the area of the washer is given by A i = π ( x i ) 2 − π ( x i −1 ) 2 = π ⎡ ⎣ x i 2 − x i −1 2 ⎤ ⎦ = π ( x i + x i −1 )( x i − x i −1 ) = π ( x i + x i −1 )Δ x . You may recall that we had an expression similar to this when we were computing volumes by shells. As we did there, we use x i * ≈ ( x i + x i −1 )/2 to approximate the average radius of the washer. We obtain A i = π ( x i + x i −1 )Δ x ≈2 πx i * Δ x . Using ρ ( x i * ) to approximate the density of the washer, we approximate the mass of the washer by m i ≈2 πx i * ρ ( x i * )Δ x . Adding up the masses of the washers, we see the mass m of the entire disk is approximated by m = ∑ i =1 n m i ≈ ∑ i =1 n 2 πx i * ρ ( x i * )Δ x . We again recognize this as a Riemann sum, and take the limit as n →∞. This gives us m = lim n →∞ ∑ i =1 n 2 πx i * ρ ( x i * )Δ x = ∫ 0 r 2 πxρ ( x ) dx . We summarize these findings in the following theorem. Theorem 6.8: Mass–Density Formula of a Circular Object Let ρ ( x ) be an integrable function representing the radial density of a disk of radius r . Then the mass of the disk is given by (6.11) m = ∫ 0 r 2 πxρ ( x ) dx .
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