Area Under A Curve

E. WHICH NUMERICAL METHOD IS MORE ACCURATE?

If you cannot view the LiveMath document below, click here to troubleshoot.

1. The 5th LiveMath document (IT4AMIntegArea5.thp) shows the same speed-time graph of the car and the calculations of the 3 numerical methods: LowerSum, UpperSum and TrapSum.

2. In Question 13, you have already decided which method is more accurate based on the graphs. But how accurate is "more accurate"? In this section, we will approach the problem from a different angle: by calculating the LowerSum, UpperSum and TrapSum for different values of n. Recall that n represents the number of rectangles or trapeziums used to approximate the area under the curve.

3. Change the value of n according to those in the table in the Worksheet and record the values of the LowerSum, UpperSum and TrapSum in the same table, leaving your answers to 4 sig. fig. where necessary. The values of n are 1, 4, 8, 15, 50, 100, 500, 1000, 5000, 10000, 30000.

Note: Don't use n > 30000. You have to wait a long time.

Question 14: As n increases, the values of the LowerSum, UpperSum and TrapSum approach the same limit. Write down the value of this limit correct to 4 sig. fig.

Question 15: This limit is the actual value of the area under the curve correct to 4 sig. fig. Which of the 3 sums approaches this limit the fastest, i.e. for the smallest value of n? State the approximate value of this smallest n (based on your table).

Question 16: Why is it so important to determine which method is more accurate for the same n? Why can't we just use a bigger n to get the same accuracy?

Hint: Do you notice how long the computer takes to calculate the 3 sums when n = 30000?

4. Double click on the Square Icon beside n = 4 in the LiveMath document above. You will see the formulae for calculating the 3 sums. Please note that the LowerSum and UpperSum basically use the same formula yx. The symbol is the summation sign. From Question 4, yx is the area of each rectangle of width x and height y. Please note that the width x is constant but the height y is different for each rectangle. So (yx) just means to sum up the areas yx of the rectangles.

5. Since the LowerSum and the UpperSum approach the limit which is the actual area (let's label the limit A), then in mathematical symbols, we write: yx A as x0 (or as n becomes bigger). We will use this later.

Continue Lesson - Section F


4.	Double click on the Square Icon beside n = 4 in the LiveMath document above. You will see the formulae for calculating the 3 sums. Please note that the LowerSum and UpperSum basically use the same formula yx. The symbol is the summation sign. From Question 4, yx is the area of each rectangle of width x and height y. Please note that the width x is constant but the height y is different for each rectangle. So (yx) just means to sum up the areas yx of the rectangles.

5.	Since the LowerSum and the UpperSum approach the limit which is the actual area (let's label the limit A), then in mathematical symbols, we write: yx A as x0 (or as n becomes bigger). We will use this later.