Riemann Sum: Area Under y = x²

The area between $y = f(x)$, the x-axis, and $x = a$, $x = b$ is the definite integral $\int_a^b f(x)\,dx$. For $f \ge 0$, this is literally the area in square units. If $f$ dips below zero, the integral subtracts that signed area.

Slide the limits $a$ and $b$ and the lab shades the region under the chosen curve and reports the integral value. Try $f(x) = x$ between $0$ and $4$: area is $\int_0^4 x\,dx = 8$ — the triangle has base 4, height 4.

It says $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F'(x) = f(x)$. Areas, distances, and accumulated quantities can be computed by reversing differentiation, no Riemann sums needed. This is the single most important idea in Class XII calculus.

Total distance from a velocity-time graph is the area under it. Pharmacokinetics computes drug exposure (AUC) by integrating concentration over time. Economists compute consumer and producer surplus as areas between supply and demand curves. The lab gives a visual anchor for all these.