Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval. Give three decimal places in your answer and explain, using a graph of f(x), what the Riemann sum in Question #1 represents.
The Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval is 28.000. This Riemann sum represents the area under the graph of f(x) from 0 to 3. The area is calculated by summing the areas of the rectangles formed by the graph of f(x) and the x-axis, with the width of each rectangle being the width of the subinterval and the height of each rectangle being the value of f(x) at the right endpoint of the subinterval.
Graph of f(x):
https://www.desmos.com/calculator/qhxhjhqhjm
To evaluate the Riemann sum for the function f(x) = x^3 - 6x, for 0 ≤ x ≤ 3 with six subintervals and right endpoints, we need to follow these steps:
Step 1: Determine the width of each subinterval.
The width of each subinterval, Δx, is calculated by dividing the total interval length (3 - 0 = 3) by the number of subintervals (6).
Δx = (3 - 0) / 6 = 0.5
Step 2: Identify the right endpoints of each subinterval.
Since we are taking the right endpoint of each subinterval, the first right endpoint will be x = 0 + Δx = 0.5, the second right endpoint will be x = 1 + Δx = 1.5, and so on. The right endpoints for the six subintervals are:
0.5, 1.5, 2.5, 3.5, 4.5, 5.5
Step 3: Evaluate f(x) at each right endpoint.
Substitute each right endpoint value into the function f(x) = x^3 - 6x to obtain the corresponding output values:
f(0.5), f(1.5), f(2.5), f(3.5), f(4.5), f(5.5)
Step 4: Calculate the sum of the products.
Multiply each output value by Δx and sum them up:
Σ[ f(xi) * Δx ], where i ranges from 1 to 6
Step 5: Round the result to three decimal places.
Finally, round the resulting sum to three decimal places.
To understand what the Riemann sum represents, let's graph the function f(x) = x^3 - 6x on the interval 0 ≤ x ≤ 3:
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The shaded area under the curve represents the Riemann sum of f(x) for our specific partition and choice of right endpoints. It approximates the definite integral of f(x) over the interval [0, 3] using rectangles, where the height of each rectangle is given by f(xi) (the function value at the right endpoint of each subinterval), and the width of each rectangle is Δx. As the number of subintervals increases and their widths approach zero, the Riemann sum converges to the definite integral of f(x) over the interval, representing the total area under the curve.
To evaluate the Riemann sum for the given function f(x) = x^3 - 6x, for 0 ≤ x ≤ 3 with six subintervals using right endpoints, we need to find the area of rectangles corresponding to each subinterval.
Step 1: Determine the width of each subinterval.
The width of each subinterval is given by (b - a) / n, where b is the upper limit, a is the lower limit, and n is the number of subintervals.
In this case, a = 0, b = 3, and n = 6.
Width of each subinterval = (3 - 0) / 6 = 0.5
Step 2: Determine the right endpoints of each subinterval.
Since we are using right endpoints, the right endpoint of each subinterval will be the starting point of the next subinterval.
The right endpoints for the six subintervals are: 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0.
Step 3: Evaluate f(x) at each right endpoint.
We substitute each right endpoint into the function to get the corresponding values.
f(0.5) = (0.5)^3 - 6(0.5) = -2.375
f(1.0) = (1.0)^3 - 6(1.0) = -5.000
f(1.5) = (1.5)^3 - 6(1.5) = -5.625
f(2.0) = (2.0)^3 - 6(2.0) = -4.000
f(2.5) = (2.5)^3 - 6(2.5) = 0.625
f(3.0) = (3.0)^3 - 6(3.0) = 9.000
Step 4: Calculate the area of each rectangle.
The area of each rectangle is given by multiplying the width of the subinterval by the corresponding function value.
Area of rectangle 1 = 0.5 * (-2.375) = -1.1875
Area of rectangle 2 = 0.5 * (-5.000) = -2.5000
Area of rectangle 3 = 0.5 * (-5.625) = -2.8125
Area of rectangle 4 = 0.5 * (-4.000) = -2.0000
Area of rectangle 5 = 0.5 * (0.625) = 0.3125
Area of rectangle 6 = 0.5 * (9.000) = 4.5000
Step 5: Sum up the areas of all rectangles.
Riemann sum = Sum of all rectangle areas = -1.1875 + (-2.5000) + (-2.8125) + (-2.0000) + 0.3125 + 4.5000 = -4.6875
The Riemann sum of -4.6875 represents an approximation of the area under the curve of the function f(x) = x^3 - 6x, for 0 ≤ x ≤ 3, using six subintervals and right endpoints as sample points. By summing up the areas of the rectangles, we obtain an estimate of the total area between the function and the x-axis within the given interval. The negative value indicates that the area is mostly below the x-axis.