Concept:
The Riemann sum is a method to approximate the value of a definite integral. The idea is to divide the interval of interest into smaller subintervals, then for each subinterval draw a rectangle whose height is determined by the function value at a chosen point (e.g., the left endpoint). The total area of all these rectangles gives an approximation of the area under the curve.

As the number of rectangles increases (and their widths decrease), this approximation becomes closer and closer to the true integral. In fact, in the limit as the rectangle width goes to zero, the Riemann sum exactly equals the definite integral.

Scenarios in this demo:
• Math Example: \(f(x) = x^2\). The sum of rectangle areas approximates the total area under the parabola between \(a\) and \(b\). This shows how the abstract idea of integration works in a purely mathematical setting.

• Real-life Example: \(v(t) = 10 + 2t - 0.2t^2\). Here the function represents the velocity of a car over time. The area under the velocity–time graph gives the total distance traveled. Since velocity is not constant, we cannot simply use "distance = velocity × time". Instead, we approximate the distance by splitting the trip into many short time intervals, assuming velocity is nearly constant in each, and summing up (velocity × time) for all intervals. The more rectangles we use, the more accurate the estimate of the car's total distance.

Approximation ≈ 0

Quick Check: Multiple Choice Question

Suppose the velocity of a car is given by \(v(t) = 10 + 2t - 0.2t^2\) (in meters/second). Which of the following best describes what the area under this curve from \(t=0\) to \(t=10\) represents?

Submit Answer