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Integral

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1. Evaluate the integral

















by making the substitution 



.

If 



, them:





 







Substituting in the integral:































































2. The traffic flow rate (cars per hour) across an intersection is   



, where  is in hours,

and  is 6 am. How many cars pass through the intersection between 6 am and 11 am?

The amount of cars (let’s call it ) that pass through the intersection in a given interval of time starting at 



and

ending at 



is given by the following integral:

















where 



 (6 am) and 







 



 



 



 (11 am).

Solving K for 



 and 



 gives:





 









 

































 













































































 







3. A company’s marginal cost function is 

















where  is the number of units. Find the total cost of the

first 25 units (from  to ).

The total cost (let’s call it ) of the first 25 units is given by the following integral:













Solving C:















 

















 





































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1. Evaluate the integral ∫ 𝒙𝟐 (𝒙𝟑 − 𝟔)𝟒𝟒 𝒅𝒙 by making the substitution 𝒖 = 𝒙𝟑 − 𝟔. If 𝑢 = 𝑥 3 − 6, them: 𝑑𝑢 = 3𝑥 2 𝑑𝑥 => 𝑑𝑥 = 𝑑𝑢 3𝑥 2 Substituting in the integral: ∫ 𝑥 2 𝑢44 𝑑𝑢 1 1 𝑢45 𝑢45 (𝑥 3 − 6)45 44 = ∫ 𝑢 𝑑𝑢 = + 𝐶 = + 𝐶 = +𝐶 3𝑥 2 3 3 45 135 135 2. The traffic flow rate (cars per hour) across an intersection is 𝒓(𝒕) = 𝟑𝟎𝟎 + 𝟕𝟎𝟎𝒕 − 𝟗𝟎𝒕𝟐 , where 𝒕 is in hours, and 𝒕 = 𝟎 is 6 am. How many cars pass through the ...
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