Use a left endpoint Riemann sum approximation with four subintervals to evaluate integral from 0 to 8 of g(x)dx
These points were given:
(0,-1)
(1,-1.5)
(2,-2.5)
(3,-3)
(4,-1.5)
(6,-0.5)
(7,-1)
(8,-1.25
you can verify your work with any of a number of fine interactive online "Riemann Sum calculator" sites
To evaluate the integral using a left endpoint Riemann sum approximation with four subintervals, we can first divide the interval from 0 to 8 into four equal subintervals:
Interval 1: [0, 2]
Interval 2: [2, 4]
Interval 3: [4, 6]
Interval 4: [6, 8]
Next, we use the left endpoint of each subinterval to approximate the value of the function at that point. The left endpoints for each subinterval are:
Endpoint 1: (0, -1)
Endpoint 2: (2, -2.5)
Endpoint 3: (4, -1.5)
Endpoint 4: (6, -0.5)
We can now calculate the individual areas for each subinterval by multiplying the width of each subinterval by the function value at the left endpoint:
Area 1: (2-0) * (-1) = 2 * (-1) = -2
Area 2: (4-2) * (-2.5) = 2 * (-2.5) = -5
Area 3: (6-4) * (-1.5) = 2 * (-1.5) = -3
Area 4: (8-6) * (-0.5) = 2 * (-0.5) = -1
Finally, we sum up the individual areas to get the approximation of the integral:
Approximation of the integral from 0 to 8 of g(x) dx ≈ -2 + (-5) + (-3) + (-1) = -11
To evaluate the integral using a left endpoint Riemann sum approximation, you need to divide the interval [0, 8] into four subintervals of equal width. The width of each subinterval can be calculated by dividing the total width of the interval by the number of subintervals.
The total width of the interval is 8 - 0 = 8.
To calculate the width of each subinterval:
Width of each subinterval = Total width / Number of subintervals
= 8 / 4
= 2
Now that we have the width of each subinterval, we can use it to determine the left endpoints of the subintervals.
The left endpoints of the subintervals are:
Endpoint 1: 0
Endpoint 2: 2
Endpoint 3: 4
Endpoint 4: 6
Next, we need to find the corresponding y-values of these left endpoints from the given points. We can use the given points to find the desired y-values.
Left Endpoint 1: (0, -1)
Left Endpoint 2: (2, -2.5)
Left Endpoint 3: (4, -1.5)
Left Endpoint 4: (6, -0.5)
To calculate the left endpoint Riemann sum approximation, we multiply the width of each subinterval with the corresponding y-value and then sum them up.
Approximation = (Endpoint 1 y-value * width) + (Endpoint 2 y-value * width) + (Endpoint 3 y-value * width) + (Endpoint 4 y-value * width)
Plugging in the values we have:
Approximation = (-1 * 2) + (-2.5 * 2) + (-1.5 * 2) + (-0.5 * 2)
= -2 + (-5) + (-3) + (-1)
= -11
Therefore, the left endpoint Riemann sum approximation with four subintervals for the integral from 0 to 8 of g(x)dx is -11.