# 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

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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, $$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx \left(-2 - \left(-3\right)\right) \left(-2\right) + \left(0 - \left(-2\right)\right) 3 + \left(3 - 0\right) \left(-1\right) + \left(5 - 3\right) 2 = 5.$$$
$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx 5$$\$A