Finding Areas under Graphs Flashcards

All 7 cards in this deck

What are the steps to estimate the area under a curve using strips of equal width?

e.g. the area under a speed–time curve from $t = 0$ to $t = 6$ using 3 strips

1. Split the region into strips of equal width: width $= 2$
2. Treat each strip as a trapezium, area $= \frac{1}{2} \times \text{width} \times (y_1 + y_2)$: e.g. $\frac{1}{2} \times 2 \times (25 + 16) = 41$
3. Add the strip areas: $41 + 25 + 13 = 79$

What is the formula for the area of one trapezium strip used to estimate the area under a curve?

e.g. a strip of width $1$ with heights $4$ and $6$

$\frac{1}{2} \times \text{width} \times (y_1 + y_2)$, where $y_1$ and $y_2$ are the two heights.

e.g. $\frac{1}{2} \times 1 \times (4 + 6) = 5$

What does the area under a speed–time (velocity–time) graph represent?

The distance travelled.

e.g. speed in m/s and time in s give a distance in metres.

True or false? On a distance–time graph, the area under the graph represents the speed.

False. Speed is given by the gradient; the area under a distance–time graph has no useful meaning.

On a graph of the rate of flow of water (litres per second) against time (seconds), what does the area under the graph represent?

The total volume of water that has flowed, in litres.

An estimate for the area under a curve is worked out using trapezium strips, and the straight top of each strip lies above the curve. Is the estimate an underestimate or an overestimate, and why?

An overestimate. The area between the trapezia and the curve is also included.

True or false? If the straight top of each trapezium strip lies below the curve, the estimate for the area under the curve is an overestimate.

False. It is an underestimate, because the area between the trapezia and the curve is not included.