Rectangles under a curve

In the picture

1. 1The curve: the graph of the function, f(x), sitting above the axis
2. 2A boundary line: one of the limits of integration, where the area starts or stops
3. 3A rectangle: width Δx times height f(x) at its left edge, one term of the sum
4. 4A blue gap: area the rectangles miss, which shrinks as they get narrower

Examples

• Estimating the distance a car travels by adding up speed × time over short time intervals
• Estimating the water poured into a tank by adding up flow rate × time for each minute

Quiz: Rectangles under a curve (Age 17)

1. 1

   How is the exact definite integral related to the rectangle sums?

   • It is the limit of the rectangle sums as each rectangle’s width tends to zero
   • It equals the area of the tallest rectangle multiplied by the number of rectangles
   • It equals the rectangle sum once you use ten equal rectangles
2. 2

   A curve rises from left to right. If each rectangle’s height is taken at its left edge, how does the rectangle sum compare with the true area?

   • It is exactly equal to the true area
   • It is larger than the true area
   • It is smaller than the true area
3. 3

   In the rectangle method, what is the area of one rectangle, that is, one term of the sum?

   • Its width Δx plus the height f(x) at its left edge
   • Its width Δx times the height f(x) at its left edge
   • The height f(x) at its left edge divided by its width Δx

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.