Area under a curve terminology: Which mathematical operation finds the accumulated area under the curve of a function over an interval?

Introduction / Context:
Integration is fundamental in physics and engineering because many quantities are cumulative: charge is the time integral of current, energy is the integral of power, and distance is the integral of velocity. In analog circuitry, op-amp integrators implement this operation over a designed bandwidth, enabling functions such as ramp generation, active filters, and control loop compensation.

Given Data / Assumptions:

• Continuous signals and standard calculus conventions.
• Ideal integrator model: output proportional to the time integral of input.
• Awareness that practical integrators include stabilizing resistors and finite limits.

Concept / Approach:
Integration computes the accumulated area under a function: for a signal v_in(t), an ideal inverting integrator produces v_out(t) = −(1/(R*C)) * ∫ v_in(t) dt over time. The area interpretation connects directly to measurement (e.g., integrating a sensor’s output to estimate position from acceleration). In active filters, integration shapes frequency response with a −20 dB/decade slope within its designed passband, complementing differentiation’s +20 dB/decade behavior.

Step-by-Step Solution:

Verification / Alternative check:
Frequency domain: integration multiplies by 1/(jω), reducing magnitude with frequency; measured Bode plots show −20 dB/decade slope. Scope traces of a square-wave input show a triangular/ramp output, visually demonstrating area accumulation.

Why Other Options Are Wrong:
Differentiation gives instantaneous slope, not area. Averaging provides mean value, not cumulative area. Linear regression estimates best-fit lines; correlation measures similarity between signals.

Common Pitfalls:
Expecting ideal behavior down to DC (practical integrators need leakage paths); ignoring op-amp saturation and input bias currents that cause drift.

Final Answer:
Integration