Left Endpoint Approximation Calculator for a Table
For the given table of values, the calculator will approximate the integral using left endpoints (the left Riemann sum), with steps shown.
Related calculator: Left Endpoint Approximation Calculator for a Function
Your Input
Approximate the integral $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx$$$ with the left endpoint approximation using the table below:
$$$x$$$ $$$-3$$$ $$$-2$$$ $$$0$$$ $$$3$$$ $$$5$$$ $$$f{\left(x \right)}$$$ $$$-2$$$ $$$3$$$ $$$-1$$$ $$$2$$$ $$$5$$$
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, $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx \left(-2 - \left(-3\right)\right) \left(-2\right) + \left(0 - \left(-2\right)\right) 3 + \left(3 - 0\right) \left(-1\right) + \left(5 - 3\right) 2 = 5.$$$
Answer
$$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx 5$$$A