Area under a curve with rectangles

In the picture

1. 1The curve of f: a continuous positive function
2. 2A rectangle: its height is the value of f at its left edge, so its area is f(xᵢ) × Δx
3. 3A blue sliver: the gap between the rectangle and the curve, which is the error of the estimate
4. 4The rectangle’s base: its width Δx = (b − a) / n

Examples

• Estimating the distance a car travels from its speed graph by adding up speed × short time intervals
• Estimating the volume of water filling a tank from its flow rate, minute by minute

Quiz: Area under a curve with rectangles (Age 18)

1. 1

   For a continuous positive function f, what does the definite integral of f between a and b represent?

   • The area of the region under the curve between a and b
   • The steepness of the curve at b
   • The highest value f reaches between a and b
2. 2

   You estimate an integral by adding up n rectangles of width Δx = (b − a) / n. What happens to the sum of their areas as n grows?

   • It gets closer and closer to the integral ∫ₐᵇ f(x) dx
   • It grows without limit, because there are more and more rectangles
   • It stays the same whatever n is, because the total width is still b − a
3. 3

   f is a rising function, and each rectangle’s height is the value of f at its left edge. What does the sum of the rectangle areas give?

   • An underestimate of the area under the curve
   • Exactly the area under the curve
   • An overestimate of the area under the curve

The answers, each with its explanation, are in the app. This lesson has left the free week, so its quiz is part of the subscription.