Use a left-endpoint Riemann sum approximation with four subintervals to evaluate ∫ (8 on top, 0 on bottom) g(x)dx
x 0 1 2 3 4 5 6 7 8
g(x) -1 -2.5 -3 -1.5 -1.25 0 -1.5 -4 -5
A. -13.5
B. 8
C. 16
D. -16
E. -8
since the width of each rectangle is 1, just take the sum
f(0)+f(1)+...+f(7) = ____
To evaluate the integral using a left-endpoint Riemann sum approximation with four subintervals, we need to calculate the width of each subinterval first.
The total width of the interval [0, 8] is 8 - 0 = 8 units.
Dividing this width by the number of subintervals (4), we get a subinterval width of 8 / 4 = 2 units.
Now, we will use the left endpoints of each subinterval to evaluate the function values and calculate the Riemann sum.
For the first subinterval [0, 2], the left endpoint is x = 0. The function value at x = 0 is g(0) = -1.
For the second subinterval [2, 4], the left endpoint is x = 2. The function value at x = 2 is g(2) = -3.
For the third subinterval [4, 6], the left endpoint is x = 4. The function value at x = 4 is g(4) = -1.25.
For the fourth subinterval [6, 8], the left endpoint is x = 6. The function value at x = 6 is g(6) = -1.5.
Now, we will calculate the Riemann sum:
Riemann sum = (width of subinterval) * (g(0) + g(2) + g(4) + g(6))
= 2 * (-1 + (-3) + (-1.25) + (-1.5))
= 2 * (-6.75)
= -13.5
Therefore, the Riemann sum approximation using four subintervals is -13.5.
The answer is A. -13.5.
To evaluate the given integral using a left-endpoint Riemann sum approximation with four subintervals, we'll need to divide the interval [0, 8] into four equal subintervals.
The width of each subinterval is:
Δx = (b - a) / n
where b is the upper limit (8), a is the lower limit (0), and n is the number of subintervals (4).
Δx = (8 - 0) / 4
Δx = 2
Now, we can calculate the left-endpoint Riemann sum by multiplying the width of each subinterval by the height of the function at the left endpoint of the subinterval. The height of the function at the left endpoint is the value of g(x) evaluated at x = a + i * Δx, where i is the index of the subinterval (0, 1, 2, ...).
For the first subinterval (i = 0):
x = 0 + 0 * 2 = 0
g(x) = g(0) = -1
Area of the first rectangle = Δx * g(x) = 2 * (-1) = -2
For the second subinterval (i = 1):
x = 0 + 1 * 2 = 2
g(x) = g(2) = -3
Area of the second rectangle = Δx * g(x) = 2 * (-3) = -6
For the third subinterval (i = 2):
x = 0 + 2 * 2 = 4
g(x) = g(4) = -1.25
Area of the third rectangle = Δx * g(x) = 2 * (-1.25) = -2.5
For the fourth subinterval (i = 3):
x = 0 + 3 * 2 = 6
g(x) = g(6) = -1.5
Area of the fourth rectangle = Δx * g(x) = 2 * (-1.5) = -3
Finally, we sum up the areas of the rectangles:
-2 + (-6) + (-2.5) + (-3) = -13.5
Therefore, the left-endpoint Riemann sum approximation of the integral ∫[0, 8] g(x) dx with four subintervals is approximately -13.5.
The correct answer is A. -13.5.