Write the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.
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To calculate the Riemann sum for the area under the graph of the function f(x) = x^4 from x = 5 to x = 7, we need to break the interval [5, 7] into smaller subintervals.
Step 1: Determine the number of subintervals.
Let's suppose we want to use n subintervals. The width of each subinterval, Δx, is given by:
Δx = (b - a) / n
where a and b are the endpoints of the interval, in this case, a = 5 and b = 7.
So, Δx = (7 - 5) / n = 2 / n.
Step 2: Identify the points within each subinterval.
We need to calculate the x-values for each subinterval. The x-values within each subinterval i can be given as:
xi = a + i * Δx
where i ranges from 0 to n-1.
In this case, a = 5, Δx = 2 / n, and for each i subinterval, xi can be found using the formula: xi = 5 + i * (2 / n).
Step 3: Calculate the y-values for each x-value.
For each x-value, we need to calculate the corresponding y-value using the function f(x) = x^4.
Step 4: Calculate the area of each rectangle.
To find the area of each rectangle, we multiply the width (Δx) with the height (the corresponding y-value).
Step 5: Sum up the areas of all rectangles.
Finally, add up the areas of all the rectangles to find the approximate area under the curve.
The Riemann sum to find the area under the graph of the function f(x) = x^4 from x = 5 to x = 7 is given by the expression:
Sum[0 to n-1] [f(xi) * Δx]
where Δx = 2 / n and xi = 5 + i * (2 / n).
Please let me know if you need any further clarification or step-by-step explanation.
To find the Riemann sum for the area under the graph of the function f(x) = x^4 from x = 5 to x = 7, you can follow these steps:
Step 1: Determine the interval between the starting and ending points.
The interval in this case is Δx = (7 - 5) = 2.
Step 2: Determine the number of subintervals.
Subintervals divide the interval into smaller intervals. You can choose the number of subintervals, denoted as n, to control the accuracy of the approximation. For simplicity, let's choose n = 4 in this example.
Step 3: Determine the width of each subinterval.
The width of each subinterval, denoted as Δx, can be calculated by dividing the interval length by the number of subintervals: Δx = 2 / 4 = 0.5.
Step 4: Determine the x-values for each subinterval.
To find the x-values for each subinterval, start with the left endpoint (5 in this case) and add Δx successively. Since there are four subintervals, you would have:
x0 = 5, x1 = x0 + Δx = 5 + 0.5 = 5.5
x2 = x1 + Δx = 5.5 + 0.5 = 6
x3 = x2 + Δx = 6 + 0.5 = 6.5
x4 = x3 + Δx = 6.5 + 0.5 = 7
Step 5: Calculate the height (f(x)) for each x-value.
For each x-value, substitute it into the function f(x) = x^4 to find the corresponding y-value:
f(x0) = (5)^4 = 625
f(x1) = (5.5)^4 ≈ 699.05
f(x2) = (6)^4 = 1296
f(x3) = (6.5)^4 ≈ 2541.06
f(x4) = (7)^4 = 2401
Step 6: Calculate the sum of the areas of the rectangles.
Sum up the areas of the rectangles formed by multiplying the width (Δx) by the height (f(x)) for each subinterval:
Area = Δx * [f(x0) + f(x1) + f(x2) + f(x3) + f(x4)]
= 0.5 * [625 + 699.05 + 1296 + 2541.06 + 2401]
≈ 0.5 * 6562.11
≈ 3281.055
Therefore, the Riemann sum for the area under the graph of the function f(x) = x^4 from x = 5 to x = 7 is approximately 3281.055 square units.
as always,
n
∑ 2/n f(5+2/n)
k=1
Since you didn't specify the endpoints or the number of intervals, you can fill in the numbers.