Final answer:
To find the evaluation points xᵢ needed in a left-end Riemann sum on the interval [1,2.6] using 4 points, we divide the interval into 4 subintervals and list the left endpoints. The values of f(xᵢ) are f(1) = 15, f(1.4) = 25.6, f(1.8) = 33.4, and f(2.2) = 51.8. The Riemann sum is computed by multiplying the width of each subinterval by the function value at each evaluation point and summing these products.
Step-by-step explanation:
A left-end Riemann sum is used to approximate the area under a curve by using left endpoints of subintervals. To find the evaluation points xᵢ needed for a left-end Riemann sum on the interval [1,2.6], we need to divide the interval into 4 subintervals. The width of each subinterval will be (2.6 - 1)/4 = 0.4. Starting with x = 1, we can list the left endpoints of the subintervals: x₁ = 1, x₂ = 1.4, x₃ = 1.8, and x₄ = 2.2.
Now, we can evaluate f(xᵢ) at each of these points. Using the function f(x) = 10x²+5, we get f(1) = 10(1)²+5 = 15, f(1.4) = 10(1.4)²+5 = 25.6, f(1.8) = 10(1.8)²+5 = 33.4, and f(2.2) = 10(2.2)²+5 = 51.8.
To compute the Riemann sum, we multiply the width of each subinterval (0.4) by the function value at each evaluation point, and then sum these products. Riemann sum = 0.4 * (15 + 25.6 + 33.4 + 51.8) = 45.2.