Science & Mathematics - 30 October 2007

Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spwed into the atmosphere are given in table 1. The time measured in seconds and the units for r(t) are tonnes (metric tons) per second.

r(0) = 2
r(1) = 10
r(2) = 24
r(3) = 36
r(4) = 46
t(5) = 54
r(6) = 60

Give upper and lower estimates for the quantity Q(6) of erupted materials after 6 seconds. Justify in complete sentences how you know you have upper and lower estimates

And use midpoint rule to estimate Q(6)

This is what I did:

I know that ∫r(t)dt = Q(t)
that must mean Q(6) is the area under the curve of r(t) from 0 to 6. So how do I get the upper bound and lower bound?

For the midpoint rule this is what i did
Q(6) = ∫r(t)dt from 0 to 6 = [(2+10)+(10+24)+(24+36)+(36+46)+(46+54)+... = 201 tonnes

??????????

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