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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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

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

(a) Give upper and lower estimates for the total quantity Q ( 6 ) of erupted materials after six seconds.

(b) Use the Midpoint Rule to estimate Q ( 6 ) .

To determine

(a)

To give:

The upper and the lower estimates for the total quantity Q(6) of erupted materials after six seconds.

Explanation

1) Concept:

Use the Net Change theorem and find the upper and the lower estimates by using Riemann sum for the right and left sample points respectively.

2) Theorem:

The integral of a rate of change is the net change.

abF'(x)dx=Fb-F(a)

3) Calculation:

The total amount of the solid material spewed into the atmosphere is given by the Net Change theorem as

Q6-Q0=06r(t)dt

since Q0=0.

From the given table, the rate r(t) is positive. Therefore, Q is an increasing function.

Hence, an upper estimate for Q(6) is R6 and the lower estimate for Q(6) is L6.

Here, n=6, a=0 and b=6.

The width is t=6-06=1.

Therefore,

R6=i=16rtit

=t[rt1+rt2+rt3+rt4+rt5+r(t6)]

=t[r1

To determine

(b)

To estimate:

The value of Q6 by using midpoint rule.

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