Area under a curve from thin rectangles

In the picture

1. 1The curve y = f(x): a continuous function that stays above the axis
2. 2The left bound x = a: where the area starts
3. 3The right bound x = b: where the area ends
4. 4One rectangle: height f(x) times width Δx, a small piece of the total area

Examples

• On a car’s speed-versus-time graph, the area under the curve between two moments is the distance travelled.
• If you know the rate water flows into a tank at every moment, the area under that rate curve gives the total volume added.

Quiz: Area under a curve from thin rectangles (Age 17)

1. 1

   For a continuous function that stays above the horizontal axis, what does the definite integral ∫ₐᵇ f(x) dx give you?

   • The exact area between the curve and the horizontal axis from a to b
   • The slope of the curve between a and b
   • The height of the curve at x = b
2. 2

   When you estimate the area under a curve by adding up rectangles, what makes the estimate become exact?

   • Letting the rectangle width Δx shrink towards 0
   • Making the rectangles wider so that fewer of them are needed
   • Giving every rectangle the same height f(a)
3. 3

   In the sum used to estimate the area, what is the area of one thin rectangle?

   • f(x) + Δx
   • f(x) × Δx
   • Δx divided by f(x)

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.