Use the upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). (Round your answers to three decimal places.)
y = sqrt of (1−x^2)
so, how many intervals? Give us your idea of a table of values for the approximation.
To approximate the area of the region using upper and lower sums, we first need to divide the interval on which the function is defined (in this case, the interval [-1, 1]) into equal subintervals.
Let's say we are given n subintervals. The width of each subinterval will be Δx = (b - a) / n, where a and b are the limits of the interval on which the function is defined (in this case, a = -1 and b = 1).
To find the upper and lower sums, we evaluate the function at specific points within each subinterval and calculate the sum of the areas of the rectangles formed by the function values and the corresponding width of the subinterval.
For the upper sum, we use the maximum value of the function within each subinterval. For the lower sum, we use the minimum value of the function within each subinterval.
Here's the step-by-step process to calculate the upper and lower sums:
1. Calculate the width of each subinterval: Δx = (1 - (-1)) / n = 2 / n.
2. Choose n points within each subinterval. One way to do this is to start from the left endpoint of each subinterval and choose equidistant points until you reach the right endpoint. For example, if n = 4, the points within each subinterval would be:
-1, -0.5, 0, 0.5 within the first subinterval
0, 0.5, 1, 1.5 within the second subinterval
1, 1.5, 2, 2.5 within the third subinterval
...
Note that the last point of one subinterval is the first point of the next subinterval.
3. Evaluate the function at each chosen point within each subinterval to find the corresponding function values. For example, if one of the chosen points is x = -0.5 within the first subinterval, evaluate y = √(1 - x^2) at x = -0.5 to find the corresponding function value within that subinterval.
4. Find the maximum and minimum function values within each subinterval. For the upper sum, take the maximum value within each subinterval. For the lower sum, take the minimum value within each subinterval.
5. Calculate the area of each rectangle within each subinterval by multiplying the function value (maximum or minimum) by the width of the subinterval.
6. Sum up the areas of all the rectangles to find the approximate area of the region using the upper sum and the lower sum. Round the answers to three decimal places, as specified in the question.
Note: As the number of subintervals increases (i.e., as n approaches infinity), the upper and lower sums will approach the actual area of the region more accurately.