Write the Riemann sum to find the area under the graph of the function f(x) = x2 from x = 1 to x = 5.
1. the summation from i equals 1 to n of the product of the quantity squared of 1 plus 5 times i over n and 4 over n
2. the limit as n goes to infinity of the summation from i equals 1 to n of the product of the quantity squared of 1 plus 4 times i over n and 4 over n
3. the summation from i equals 1 to n of the product of the quantity squared of 4 times i over n and 4 over n
4. the limit as n goes to infinity of the summation from i equals 1 to n of the product of i over n quantity squared and 4 over n
well, the right-hand sum is
n
∑ f(1+i(4/n))(4/n)
i=1
there's also the left sum and the midpoint sum
The correct Riemann sum to find the area under the graph of the function f(x) = x^2 from x = 1 to x = 5 is option 1:
The summation from i equals 1 to n of the product of the quantity squared of 1 plus 5 times i over n and 4 over n.
To understand how to arrive at this answer, let's break it down step by step:
1. Divide the interval [1, 5] into n subintervals, each with width Δx = (5 - 1) / n = 4/n.
2. Choose a representative value (xi) for each subinterval. In this case, we will use the right endpoint of each subinterval.
The right endpoints of the subintervals can be calculated as xi = 1 + (5 - 1) * i / n, where i ranges from 1 to n.
3. Calculate the height of the rectangle for each subinterval. Since f(x) = x^2, the height will be f(xi) = (1 + 5i/n)^2.
4. Multiply the width and the height of each rectangle: Δx * f(xi) = (4/n) * (1 + 5i/n)^2.
5. Take the summation of all the rectangle areas from i = 1 to n: Σ[(4/n) * (1 + 5i/n)^2], where Σ denotes the summation symbol.
Therefore, the correct Riemann sum is the summation from i equals 1 to n of the product of the quantity squared of 1 plus 5 times i over n and 4 over n.
The correct answer is:
4. The limit as n goes to infinity of the summation from i equals 1 to n of the product of i over n squared and 4 over n.
Explanation:
To find the Riemann sum for the area under the graph of the function f(x) = x^2 from x = 1 to x = 5, we divide the interval [1, 5] into n equal subintervals. Each subinterval will have a width of (5 - 1)/n = 4/n.
We then choose a sample point in each subinterval, denoted as xi, and evaluate the function at that point to get f(xi). In this case, the function is f(x) = x^2, so f(xi) = (xi)^2.
Finally, we multiply the function value f(xi) by the width of the subinterval to get the area of the corresponding rectangle. Thus, the area of each rectangle is (4/n)(xi^2).
To find the total area under the graph, we take the summation of the areas of all the rectangles from i = 1 to i = n, and then take the limit as n approaches infinity.
Therefore, the correct Riemann sum is the limit as n goes to infinity of the summation from i equals 1 to n of the product of i over n squared and 4 over n.