Estimate �ç 6 0 6−x^2 dx using 3 right rectangles
To estimate the given integral using right rectangles, we need to divide the interval of integration into equal subintervals and approximate the area under the curve by summing the areas of the rectangles.
In this case, we are given the function f(x) = 606 - x^2 and we need to estimate the integral ∫(606 - x^2) dx over the interval [a, b].
Step 1: Determine the interval [a, b]
Since the question does not specify the interval, we'll assume it to be from a = 0 to b = 6.
Step 2: Divide the interval into subintervals
We are asked to use 3 right rectangles, so we'll divide the interval [0, 6] into 3 equal subintervals. The width of each subinterval will be (b - a) / n = (6 - 0) / 3 = 2.
The three subintervals will be:
[0, 2] - first subinterval
[2, 4] - second subinterval
[4, 6] - third subinterval
Step 3: Approximate the area with right rectangles
We'll calculate the height of each rectangle by evaluating the function at the right endpoint of each subinterval.
For the first subinterval [0, 2], the right endpoint is x = 2.
Height of the rectangle for this subinterval: f(2) = 606 - (2^2) = 606 - 4 = 602.
For the second subinterval [2, 4], the right endpoint is x = 4.
Height of the rectangle for this subinterval: f(4) = 606 - (4^2) = 606 - 16 = 590.
For the third subinterval [4, 6], the right endpoint is x = 6.
Height of the rectangle for this subinterval: f(6) = 606 - (6^2) = 606 - 36 = 570.
Step 4: Calculate the approximate area under the curve
Now, we'll calculate the area of each rectangle and sum them up:
Area of the first rectangle = (width) * (height) = 2 * 602 = 1204.
Area of the second rectangle = (width) * (height) = 2 * 590 = 1180.
Area of the third rectangle = (width) * (height) = 2 * 570 = 1140.
Summing up the areas of the rectangles: 1204 + 1180 + 1140 = 3524.
Therefore, the estimated value of the integral ∫(606 - x^2) dx over the interval [0, 6] using 3 right rectangles is approximately 3524.