The marginal cost of manufacturing x yards of a certain fabric is: C'(x) = 3 - 0.01*x + 0.000009*x^2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.

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The marginal cost of manufacturing x yards of a certain fabric is:

C'(x) = 3 - 0.01x + 0.000009*x^2.

The increase in cost if the production goes from x = 2000 to x = 4000 is the definite integral:

`int_(2000)^4000 3 - 0.01x + 0.000009*x^2 dx`

=> `3x - 0.01x^2/2...

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The marginal cost of manufacturing x yards of a certain fabric is:

C'(x) = 3 - 0.01x + 0.000009*x^2.

The increase in cost if the production goes from x = 2000 to x = 4000 is the definite integral:

`int_(2000)^4000 3 - 0.01x + 0.000009*x^2 dx`

=> `3x - 0.01x^2/2 + 0.000009*x^3/3` between x = 2000 and x = 4000

=> `3(4000-2000) - 0.005(4000^2 - 2000^2) + 0.000003*(4000^3 - 2000^3)`

=> `3*2000 - 0.005*12*10^6 + 0.000003*5.6*10^10`

=> 6000 - 60000 + 168000

=> 114000

The increase in cost is $114000

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