The rectangles in the graph illustrates a left endpoint Riemann sum for f(x)=−(x^2/4)+2x on the interval [3,7].
The value of this left endpoint Riemann sum is?
The rectangles in the graph illustrates a right endpoint Riemann sum for f(x)=−(x^2/4)+2x on the interval [3,7].
The value of this right endpoint Riemann sum is
To find the value of a Riemann sum, you need to calculate the sum of the areas of the rectangles that approximate the area under the curve.
For a left endpoint Riemann sum, the height of each rectangle is determined by evaluating the function at the left endpoint of each subinterval. In this case, we have the function f(x) = -(x^2/4) + 2x and the interval [3,7].
To calculate the width of each subinterval, you can use the formula (b - a) / n, where 'b' is the upper bound of the interval, 'a' is the lower bound of the interval, and 'n' is the number of subintervals (in this case, the number of rectangles).
In this case, b = 7, a = 3, and let's assume we have 4 rectangles, so n = 4.
Therefore, the width of each subinterval is (7 - 3) / 4 = 1.
Now, we can calculate the height of each rectangle by evaluating the function at the left endpoint of each subinterval:
For the left endpoint of the first subinterval (3), f(3) = -(3^2/4) + 2(3) = -2.25 + 6 = 3.75.
For the left endpoint of the second subinterval (4), f(4) = -(4^2/4) + 2(4) = -4 + 8 = 4.
For the left endpoint of the third subinterval (5), f(5) = -(5^2/4) + 2(5) = -6.25 + 10 = 3.75.
For the left endpoint of the fourth subinterval (6), f(6) = -(6^2/4) + 2(6) = -9 + 12 = 3.
With the width and height of each rectangle, we can calculate the area of each rectangle by multiplying the width and height.
For the left endpoint Riemann sum, the value is the sum of the areas of all the rectangles. So, add up the areas of all the rectangles and you will get the value of the left endpoint Riemann sum.
Similarly, you can follow the same steps to calculate the value of the right endpoint Riemann sum, but in this case, you would evaluate the function at the right endpoint of each subinterval instead of the left endpoint.