Express the integral as a limit of Riemann sums. Do not evaluate the limit.
∫_1^3 x/(2+x^5)
Ans:
lim(n→∞)∑_(i=1)^n ?
To express the integral ∫₁³ x/(2 + x⁵) as a limit of Riemann sums, we need to consider the partitions of the interval [1, 3] into n equal subintervals.
First, we calculate the width of each subinterval. Let Δx be the width of each subinterval, which is given by
Δx = (b - a) / n
where a is the lower limit of integration (1 in this case) and b is the upper limit of integration (3 in this case).
Next, we choose representative points within each subinterval. Let xᵢ be the representative point for the i-th subinterval. We can choose xᵢ as the right endpoint, left endpoint, or any point within the subinterval. For simplicity, let's choose xᵢ as the right endpoint of each subinterval.
Now, we can express the integral as a limit of Riemann sums using the formula:
∫₁³ x/(2 + x⁵) = lim(n→∞) ∑ᵢ f(xᵢ) Δx
where f(x) is the function x/(2 + x⁵), xᵢ is the right endpoint of the i-th subinterval, and Δx is the width of each subinterval.
Therefore, the expression becomes:
lim(n→∞) ∑ᵢ (xᵢ / (2 + xᵢ⁵)) Δx
where xᵢ = a + iΔx, i varies from 0 to n-1, and Δx = (3 - 1) / n.
Please note that this expression represents the limit of Riemann sums and is not the final numerical value of the integral. To evaluate the integral, we would need to take the limit as n approaches infinity and perform the necessary calculations.