Final answer:
The estimated value of the integral from 0 to 4 of the function 1/(x² + 1) using the partitions 0, 2, and 4 is 1.2, obtained using the Midpoint Rule.
Step-by-step explanation:
The student has been asked to estimate the integral of the function 1/(x² + 1) from 0 to 4 using the partitions 0, 2, and 4. This problem involves estimating the area under the curve of the given function within the specified interval. Let's perform the estimation using the midpoint rule for simplicity:
Divide the interval [0, 4] into two subintervals: [0, 2] and [2, 4].
Find the midpoint of each subinterval: m1 = 1 and m2 = 3.
Calculate the function values at these midpoints: f(m1) = 1/(1² + 1) = 0.5, f(m2) = 1/(3² + 1) = 1/10.
Multiply each function value by the width of the subintervals (which is 2): A1 = 2 * 0.5 = 1, A2 = 2 * 1/10 = 1/5.
Add the areas of the subintervals to get the total estimated area: A = A1 + A2 = 1 + 1/5 = 1.2.
Thus, the estimated value of the integral ∫(4 to 0) 1/(x² + 1) dx using the partitions 0, 2, and 4 is 1.2.