$\"\"$

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$\"\"$ is erupting rate of volcano in tonnes per second.

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(a)

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Find the upper and lower estimates for the total quantity $\"\"$ after 6 seconds:

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Upper estimtes refers to the right end point method.

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Number of intervals are $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Width of the each interval : $\"\"$ .

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Right end points are $\"\"$ and $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Total quantity of erupted material using upper estimates is $\"\"$ .

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Lower estimtes refers to the left end point method.

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Left end points are $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Total quantity of erupted material using lower estimates is $\"\"$ .

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$\"\"$

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(b) Use mid point rule to estimate total quantity $\"\"$ :

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Consider each interval as $\"\"$ seconds.

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Mid points are $\"\"$ and $\"\"$ .

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Number of intervals are $\"\"$ .

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Width of the each interval : $\"\"$ .

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Mid point esimation:

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$\"\"$

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$\"\"$

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$\"\"$ .

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Total quantity of erupted material using midpoint estimates is $\"\"$ .

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$\"\"$

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(a) Upper estimtes for the total quantity $\"\"$ of erupted materials after 6 seconds is $\"\"$ tonnes.

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Lower estimtes for the total quantity $\"\"$ of erupted materials after 6 seconds is $\"\"$ tonnes.

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(b) Mid point estimtes for the total quantity $\"\"$ of erupted materials after 6 seconds is $\"\"$ tonnes.