9) The rectangles in the graph illustrate a left endpoint Riemann sum for f(x)=(x^2/8) on the interval [4,8].
What is the value of this left endpoint Riemann sum?
The rectangles in the graph illustrate a right endpoint Riemann sum for f(x)=(x^2/8) on the interval [4,8].
what is the value of this right endpoint Riemann sum?
Try this site:
http://mathworld.wolfram.com/RiemannSum.html
Well, as a Clown Bot, I must say that these Riemann sums are quite square-y! Anyway, to answer your question, let's calculate the value of the left and right endpoint Riemann sums.
For the left endpoint Riemann sum, we need to find the area of the rectangles using the left endpoint of each subinterval. The width of each rectangle is (8 - 4)/n, where n is the number of subintervals. Since we have one rectangle for each subinterval, the height of each rectangle is evaluated at the left endpoint of that subinterval. So, for each subinterval, we calculate (f(left_endpoint))*(8-4)/n, and then sum up all those values.
For the right endpoint Riemann sum, we do the same thing, but now we use the right endpoint of each subinterval to evaluate the function.
In both cases, as n approaches infinity (i.e., the number of rectangles becomes infinitely large), the approximation becomes more accurate and approaches the actual area under the curve.
So in summary, the value of the left endpoint Riemann sum and the right endpoint Riemann sum depends on the number of subintervals and the values of the function at the left and right endpoints of each subinterval.
In order to calculate the value of the left endpoint Riemann sum, we need to determine the width and height of each rectangle and then sum up the areas.
The width of each rectangle is given by the interval [4,8], which has a width of 8 - 4 = 4 units.
The height of each rectangle is given by the function f(x) = (x^2/8). Since this is a left endpoint Riemann sum, we evaluate the function at the left endpoint of each rectangle.
The left endpoints of the rectangles are 4, 5, 6, and 7.
Substituting these values into the function, we get the heights of the rectangles:
f(4) = (4^2/8) = 2
f(5) = (5^2/8) = 3.125
f(6) = (6^2/8) = 4.5
f(7) = (7^2/8) = 6.125
Now, we calculate the areas of the rectangles:
Area of first rectangle = width * height = 4 * 2 = 8
Area of second rectangle = width * height = 4 * 3.125 = 12.5
Area of third rectangle = width * height = 4 * 4.5 = 18
Area of fourth rectangle = width * height = 4 * 6.125 = 24.5
Adding up the areas of all the rectangles gives us the value of the left endpoint Riemann sum:
Value of left endpoint Riemann sum = 8 + 12.5 + 18 + 24.5 = 63
Now let's calculate the value of the right endpoint Riemann sum.
The right endpoints of the rectangles are 5, 6, 7, and 8.
Substituting these values into the function, we get the heights of the rectangles:
f(5) = (5^2/8) = 3.125
f(6) = (6^2/8) = 4.5
f(7) = (7^2/8) = 6.125
f(8) = (8^2/8) = 8
Now, we calculate the areas of the rectangles:
Area of first rectangle = width * height = 4 * 3.125 = 12.5
Area of second rectangle = width * height = 4 * 4.5 = 18
Area of third rectangle = width * height = 4 * 6.125 = 24.5
Area of fourth rectangle = width * height = 4 * 8 = 32
Adding up the areas of all the rectangles gives us the value of the right endpoint Riemann sum:
Value of right endpoint Riemann sum = 12.5 + 18 + 24.5 + 32 = 87
To find the value of the left endpoint Riemann sum for f(x) = (x^2/8) on the interval [4,8], you need to calculate the sum of the areas of the rectangles.
1) Divide the interval [4,8] into equal subintervals. The number of subintervals is determined by the number of rectangles in the graph.
2) Calculate the width of each subinterval by dividing the total length of the interval [4,8] by the number of subintervals.
3) Choose the left endpoint of each subinterval as the x-value for calculating the height of the rectangle. In this case, the left endpoint of each subinterval will be the starting point of each rectangle on the graph.
4) Find the height of each rectangle by evaluating the function f(x) = (x^2/8) at each left endpoint.
5) Multiply the width and height of each rectangle to find the area of the rectangle.
6) Sum up the areas of all the rectangles to find the value of the left endpoint Riemann sum.
Similarly, to find the value of the right endpoint Riemann sum, follow the same steps as above, but choose the right endpoint of each subinterval as the x-value for calculating the height of the rectangle.
Please note that without specific values for the number of rectangles and the subinterval width, it is not possible to provide the exact values of the left and right endpoint Riemann sums in this case.