Question

rod has length 5 meters. At a distance x meters from its left end, the density of the rod is given by 8(x) = 3 + 7x g/m. (a) Complete the Riemann sum for the total mass of the rod (use Dx in place of Ax): mass = Σ (b) Convert the Riemann sum to an integral and find the exact mass. mass = (include units)

Answer 1

The total mass of the rod is calculated by summing the product of density and length over the rod using Riemann sums and then evaluating the integral to find the exact mass, which is 102.5 grams.

Step-by-step explanation:

The question asks to find the total mass of a rod using Riemann sums and then to convert that sum into an integral to find the exact mass. The density δ(x) of the rod is a function of the distance x from one end and is given by δ(x) = 3 + 7x grams per meter. To create a Riemann sum for the total mass, you would sum up the product of the density and the small lengths Δx over the entire length of the rod:

mass = Σ (3 + 7x_i) Δx

To convert this to an integral, we take the limit as Δx approaches zero:

mass = ∫_0^5 (3 + 7x) dx

The exact mass is found by evaluating this integral:

mass = ∫_0^5 (3 + 7x) dx = [3x + (7/2)x^2]_0^5 = (3⋅ 5) + (7/2 ⋅ 25) = 15 + 87.5 = 102.5 grams