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Integration maths a level help

The speed v in ms-1 is given at time t seconds in the following table
T= 0 v= 0 t= 10 v=5.0 t = 20 v=6.7 t = 30 v=8.2 t = 40 v=9.5 t = 50 v=10.6 t = 60 v=11.6
the distance that the train travelled is given by the area under the graph of the speed (vertical axis) against time (horizontal axis). Estimate the distance the train travels in this 1 minute period. This was my working:
I used that limit symbol with 60 at the top and 0 at the bottom, and found the gradient of the line to be 11.6/60 = 29/150. I integrated this to 29/150 t^2 all divided by 2. I substituted 60 into t and got 348. I didn't substitute 0 as I knew this would just equal 0. However the answer is 458 m, which is not my answer

Reply 1

jofuone

just imagine you've drawn the graph, then you just have 6 trapeziums - from 0 to 10 seconds(which is actually just a triangle), 10 to 20, 20 to 30 etc
the area of each trapezium is 10 * (speed at start+speed at end)/2 add them up and hey presto 458 metres.

if you now draw a straight line from 0,0 to 60,11.6 you'll see that there's a large amount of space between your straight line and the "curve" of the data points you've been given, that explains the difference between your 348 and the 458.

Reply 2

ghostwalker

Here's an image of the area you worked out - in green - and the actual data - the line above it (i've assumed it's linear between data points).

Reply 3

scphmapd

Original post by ghostwalker

Here's an image of the area you worked out - in green - and the actual data - the line above it (i've assumed it's linear between data points).

If this is just an estimate, then you want to change the green triangle so that it includes some of the area outside of the graph as well as some of the area inside the graph. This will give you a closer approximation, but is not totally accurate.

Reply 4

ghostwalker

Original post by scphmapd

The green triangle is the area the OP actually worked out.

The piecewise linear line represents the given data points.

Just done to highlight the difference to the OP.