Area Under A Curve

A. SPEED-TIME GRAPH

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1. The 1st LiveMath document (IT4AMIntegArea1.thp) shows the speed-time graph of a car travelling at a speed of 15 m/s for 4 s.

Question 1: What is the distance travelled?

Distance = Speed x Time = __ x __ = ___ m

Or Distance = Area under the curve

= __ x __ = ___ m

Observe that there are ___ ways to calculate the distance if the speed is constant.

2. Suppose the car travels at a speed v = 4t for 4s. Note that the speed is not constant. Change the equation of the curve in the document from v = 15 to v = 4t and observe the change.

Question 2: What is the distance travelled?

Distance = Area under the curve

= 1/2 x __ x __ = ___ m

3. Suppose the car travels at a speed v = t² for 4s. Change the equation of the curve in the document from v = 4t to v = t² and observe the change. To key in t², type t^2.

4. What is the distance travelled? How do you find the area under a curve which is not a straight line?

5. First, we try to approximate the area using a series of rectangles of width (b–a)/n and length y. There are 2 ways to do this: the Lower Riemann Sum where all the rectangles are under the curve; the Upper Riemann Sum where all the rectangles cover the curve.

Continue Lesson - Section B


3.	Suppose the car travels at a speed v = t² for 4s. Change the equation of the curve in the document from v = 4t to v = t² and observe the change. To key in t², type t^2.

4.	What is the distance travelled? How do you find the area under a curve which is not a straight line?

5.	First, we try to approximate the area using a series of rectangles of width (b–a)/n and length y. There are 2 ways to do this: the Lower Riemann Sum where all the rectangles are under the curve; the Upper Riemann Sum where all the rectangles cover the curve.