What is the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.
You did not say how many rectangles. Here it is for ten:
http://mathworld.wolfram.com/RiemannSum.html
To find the Riemann sum for the area under the graph of a function, we need to divide the interval [5, 7] into smaller subintervals and then calculate the sum of the areas of rectangles formed by the function and the width of each subinterval.
Here's how you can find the Riemann sum for the function f(x) = x^4 from x = 5 to x = 7:
Step 1: Determine the number of subintervals, denoted by n, that you want to divide the interval [5, 7] into. The larger the value of n, the more accurate the approximation will be, but it will also require more calculations.
Step 2: Calculate the width of each subinterval, denoted by Δx, by taking the difference between the endpoints of the interval and dividing it by the number of subintervals:
Δx = (7 - 5) / n
Step 3: Choose the points within each subinterval at which you will evaluate the function. The most commonly used choices are the left endpoints, right endpoints, or midpoints of each subinterval. Let's use the right endpoints for this example.
Step 4: Evaluate the function at the chosen points within each subinterval. In this case, we will evaluate f(x) = x^4 at the right endpoint of each subinterval.
Step 5: Multiply the function values obtained in Step 4 by the width of each subinterval (Δx) to get the area of each rectangle.
Step 6: Sum up the areas of all the rectangles to get the Riemann sum, denoted by S_n. This is the approximate area under the graph of the function.
In this specific example, let's assume we want to divide the interval [5, 7] into 4 subintervals (n = 4) using the right endpoints of each subinterval.
Δx = (7 - 5) / 4 = 0.5
Now, evaluate the function f(x) = x^4 at the right endpoints of each subinterval:
For the first subinterval: x = 5 + Δx = 5 + 0.5 = 5.5
f(5.5) = (5.5)^4 = 915.0625
For the second subinterval: x = 5.5 + Δx = 5.5 + 0.5 = 6
f(6) = (6)^4 = 1296
For the third subinterval: x = 6 + Δx = 6 + 0.5 = 6.5
f(6.5) = (6.5)^4 = 1785.0625
For the fourth subinterval: x = 6.5 + Δx = 6.5 + 0.5 = 7
f(7) = (7)^4 = 2401
Now, multiply each function value by the width of the subintervals:
Rectangle 1 area = f(5.5) * Δx = 915.0625 * 0.5 = 457.53125
Rectangle 2 area = f(6) * Δx = 1296 * 0.5 = 648
Rectangle 3 area = f(6.5) * Δx = 1785.0625 * 0.5 = 892.53125
Rectangle 4 area = f(7) * Δx = 2401 * 0.5 = 1200.5
Finally, sum up the areas of all the rectangles:
S_n = Rectangle 1 area + Rectangle 2 area + Rectangle 3 area + Rectangle 4 area
= 457.53125 + 648 + 892.53125 + 1200.5
= 3198.5625
Therefore, the Riemann sum to find the area under the graph of the function f(x) = x^4 from x = 5 to x = 7, using 4 subintervals and the right endpoints, is approximately 3198.5625.
To find the Riemann sum for the area under the graph of the function f(x) = x^4 from x = 5 to x = 7, we can start by dividing the interval [5, 7] into smaller subintervals.
Step 1: Determine the number of subintervals.
We can choose any positive integer n to determine the number of subintervals. Let's assume n = 4 for this example.
Step 2: Calculate the width of each subinterval.
We can calculate the width of each subinterval by dividing the total interval width by the number of subintervals.
Interval width = 7 - 5 = 2
Subinterval width = Interval width / number of subintervals = 2 / 4 = 0.5
Step 3: Determine the x-values for each subinterval.
We can determine the x-values for each subinterval by starting with the left endpoint of the interval (x = 5) and adding the subinterval width successively.
Subinterval 1: x = 5
Subinterval 2: x = 5 + 0.5 = 5.5
Subinterval 3: x = 5.5 + 0.5 = 6
Subinterval 4: x = 6 + 0.5 = 6.5
Step 4: Evaluate the function at each x-value.
Evaluate the function f(x) = x^4 at each x-value calculated in step 3.
For Subinterval 1: f(5) = 5^4 = 625
For Subinterval 2: f(5.5) = 5.5^4 = 915.0625
For Subinterval 3: f(6) = 6^4 = 1296
For Subinterval 4: f(6.5) = 6.5^4 = 1891.5625
Step 5: Calculate the area of each subinterval.
To find the area of each subinterval, multiply the subinterval width by the corresponding function value obtained in step 4.
For Subinterval 1: Area 1 = 0.5 * 625 = 312.5
For Subinterval 2: Area 2 = 0.5 * 915.0625 = 457.53125
For Subinterval 3: Area 3 = 0.5 * 1296 = 648
For Subinterval 4: Area 4 = 0.5 * 1891.5625 = 945.78125
Step 6: Sum up the areas of all subintervals.
Calculate the sum of all the subinterval areas obtained in step 5.
Total Area = Area 1 + Area 2 + Area 3 + Area 4
= 312.5 + 457.53125 + 648 + 945.78125
= 2363.8125
Therefore, the Riemann sum for the area under the graph of the function f(x) = x^4 from x = 5 to x = 7, with 4 subintervals, is approximately 2363.8125 square units.