The rectangles in the graph below illustrate a left endpoint Riemann sum for \( f(x)=\frac{x^{2}}{10} \) on the interval \( [3,7] \). The value of this left endpoint Riemann sum is Riemann sum is an underestimate of the area of the region enclosed by \( y=f(x) \), the x -axis, and the vertical lines \( \mathrm{x}=3 \) and \( \mathrm{x}=7 \).

Did you know that Riemann sums were developed by German mathematician Bernhard Riemann in the 19th century? He laid the groundwork for integral calculus by introducing these sums, allowing mathematicians to approximate the area under curves. It was a pivotal development that greatly influenced our understanding of calculus and paved the way for more advanced concepts like limits and continuity! In practical terms, Riemann sums can be incredibly useful in computer graphics and numerical analysis. They allow us to approximate areas and volumes, which is essential when designing simulations or rendering scenes. By breaking complex shapes into simpler rectangles, applications can effectively compute integrals—even in asynchronous environments like video games—where real-time calculations are crucial. So next time you see a smooth curve in a game, remember that Riemann sums might be at work behind the scenes!