Computational Mathematics & Manifolds

f(x) = \sin(x) \cdot e^{-0.15 x}

Instantaneous Derivativef'(x₀) = -0.4093

Riemann Sum (S_N)1.3958

Definite Integral (Exact)∫ f(x)dx = 1.3898

Approximation Error0.43 %

Derivative Probe Point (x₀)2.00

Lower Limit (a)0.00

Upper Limit (b)4.00

Riemann Sum Partition Rule:

Partitions (N)12

N = 2N = 60

As the partition count $N \to \infty$ (mesh size $\Delta x \to 0$ ), the Riemann summation strictly converges to the true definite integral:

\lim_{N \to \infty} \sum_{i=1}^N f(x_i^*) \Delta x = \int_a^b f(x)\,dx