Given the table below for selected values of f(x), use 6 left rectangles to estimate the value of : integral from 1 to 10 f(x)dx
x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
Numerical answers expected:
70
There are lots of good online Riemann Sum calculators, such as this one at
https://www.emathhelp.net/calculators/calculus-2/riemann-sum-calculator/
I know that "a" is 1 and "b" is 10 and "n" is equal 6 but
-how would it be the function to put in the calculator
- and would be Left Riemann Sum, right?
To estimate the value of the integral using left rectangles, we can use the following steps:
Step 1: Find the width of each rectangle.
To estimate the integral using left rectangles, we need to divide the interval [1, 10] into equal subintervals. Since we have 6 rectangles, the width of each rectangle will be Δx = (10 - 1) / 6 = 1.5.
Step 2: Identify the height of each rectangle.
For left rectangles, we use the value of f(x) at the left endpoint of each subinterval as the height of the rectangle.
The left endpoints for the given table are:
1, 3, 4, 6, 7, 9
The corresponding heights are:
f(1) = 4, f(3) = 8, f(4) = 6, f(6) = 10, f(7) = 10, f(9) = 12
Step 3: Calculate the area of each rectangle.
The area of each rectangle is A = height * width.
Rectangle 1: A1 = f(1) * Δx = 4 * 1.5 = 6
Rectangle 2: A2 = f(3) * Δx = 8 * 1.5 = 12
Rectangle 3: A3 = f(4) * Δx = 6 * 1.5 = 9
Rectangle 4: A4 = f(6) * Δx = 10 * 1.5 = 15
Rectangle 5: A5 = f(7) * Δx = 10 * 1.5 = 15
Rectangle 6: A6 = f(9) * Δx = 12 * 1.5 = 18
Step 4: Add up the areas of all rectangles.
The estimate of the integral is obtained by summing up the areas of all rectangles:
Estimate = A1 + A2 + A3 + A4 + A5 + A6 = 6 + 12 + 9 + 15 + 15 + 18 = 75.
Therefore, the estimate for the value of the integral of f(x) from 1 to 10 using 6 left rectangles is 75.