The approximation to the area under the graph of the function f(x)= 1/(1+x^2)
on the interval [-1,1], using four rectangles and the midpoint rule is
To approximate the area under the graph of the function f(x) = 1/(1+x^2) on the interval [-1,1] using four rectangles and the midpoint rule, follow these steps:
1. Divide the interval [-1,1] into four equal subintervals. Since there are four rectangles, each subinterval will have a width of (1 - (-1)) / 4 = 1/2.
2. For each subinterval, find the midpoint. The midpoint of each subinterval can be found by taking the average of its endpoints. For example, for the first subinterval, the midpoint is (-1 + (-1 + 1/2))/2 = -3/4.
3. Evaluate the function f(x) = 1/(1+x^2) at each midpoint. In this case, plug in the value of each midpoint into the function to get the corresponding function values. For example, f(-3/4) = 1/(1+(-3/4)^2) = 1/(1+9/16) = 16/25.
4. Multiply each function value by the width of the subinterval. Since the width is 1/2 for each subinterval, multiply each function value by 1/2 to get the area of each rectangle. For example, for the first rectangle, the area is (16/25) * (1/2) = 8/25.
5. Sum up the areas of all the rectangles to get the total approximation of the area under the graph. In this case, add up the areas of all four rectangles.
The final approximation to the area under the graph of f(x) = 1/(1+x^2) on the interval [-1,1], using four rectangles and the midpoint rule, is the sum of the areas of all four rectangles.