Riemann Sum Calculator for a Table
Approximate an integral (given by a table of values) using the Riemann sum step by step
For the given table of values, the calculator will approximate the definite integral using the Riemann sum and the sample points of your choice: left endpoints, right endpoints, midpoints, and trapezoids.
Related calculator: Riemann Sum Calculator for a Function
Your Input
Approximate the integral $$$\int\limits_{0}^{8} f{\left(x \right)}\, dx$$$ with the left Riemann sum using the table below:
| $$$x$$$ | $$$0$$$ | $$$2$$$ | $$$4$$$ | $$$6$$$ | $$$8$$$ |
| $$$f{\left(x \right)}$$$ | $$$1$$$ | $$$-2$$$ | $$$5$$$ | $$$0$$$ | $$$7$$$ |
Solution
The left Riemann sum approximates the integral using left endpoints: $$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{n - 1} \left(x_{i+1} - x_{i}\right) f{\left(x_{i} \right)}$$$, where $$$n$$$ is the number of points.
Therefore, $$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \left(2 - 0\right) 1 + \left(4 - 2\right) \left(-2\right) + \left(6 - 4\right) 5 + \left(8 - 6\right) 0 = 8$$$.
Answer
$$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx 8$$$A