Calculate the indicated Riemann Sums for the function g(x)=4-x^2
a)one rectangle
b)two rectangles
You have to give beginning and end.
To calculate the indicated Riemann Sums for a function, we need to divide the interval of the function into subintervals and approximate the area under the curve using rectangles.
The formula for a Riemann sum with n rectangles is:
R_n = ∑[i=1 to n] f(x_i) * Δx
where R_n is the Riemann sum, f(x_i) is the function evaluated at a specific x-value in each subinterval, and Δx is the width of each subinterval.
a) One Rectangle:
To find the area under the curve using one rectangle, we only need to evaluate the function at one point within the interval.
First, let's determine the interval. If the interval is not provided, we'll assume it's from a to b, where a and b are the bounds of the interval.
Let's say the interval is [a, b]. For one rectangle, we can simply choose the midpoint of the interval as x-value. So, the specific x-value for the one rectangle is (a+b)/2.
The width of the rectangle, Δx, is equal to the length of the interval, which is (b - a).
Now, substitute the values into the Riemann sum formula:
R_1 = f((a+b)/2) * (b - a)
R_1 = g((a+b)/2) * (b - a)
R_1 = (4 - ((a+b)/2)^2) * (b - a)
b) Two Rectangles:
To find the area under the curve using two rectangles, we need to divide the interval into two equal subintervals.
Again, let the interval be [a, b].
To divide the interval into two equal parts, we need to find the midpoint, let's call it c:
c = (a + b) / 2
Now, we can calculate the width of each subinterval, Δx:
Δx = (b - a) / 2
To calculate the Riemann sum, we evaluate the function at two specific x-values within the subintervals.
For the first rectangle, the x-value is (a+c)/2, and for the second rectangle, it is (c+b)/2.
Let's substitute the values into the Riemann sum formula:
R_2 = f((a+c)/2) * Δx + f((c+b)/2) * Δx
R_2 = g((a+c)/2) * Δx + g((c+b)/2) * Δx
R_2 = (4 - ((a+c)/2)^2) * Δx + (4 - ((c+b)/2)^2) * Δx
By following these steps, we can calculate the indicated Riemann Sums for any function. Just make sure to choose the appropriate number of rectangles and evaluate the function at the correct x-values within each subinterval.