Integrals Probelms

1. Evaluate the following integrals

A. f^2 0 (3x^3 –x^2 +2) dx this is (f) with subscript of 0 and (f) to the power of 2.

B. f^1 0 (e^(2x)-x^2) dx this is (f) to the power of 1 and (f) with a subscript of 0

C. f (1/x+3)(dx)

2. In a test run, a new train travels along a straight-line track. Data obtained 
From the speedometer, indicate that the velocity of the train at any time t can be 
Described by the velocity function 

V (t) =8t (0≤t≤30)

a. Find the position function of the train. 
b. Find the position after 3 seconds. (Note: the train starts from the beginning 
of the track so when t = 0, the integration constant, C = 0.) 

3. The current circulation of a particular magazine is 3,000 copies per week. The 
Editor projects a growth rate of 

G (t) = 4+5t^2/3

Copies per week after t weeks. 

a. Find the circulation function based on this projection. 
b. Find the circulation in 2 years. 

4. A company manufactures widgets. The daily marginal cost to produce x 
Widgets is found to be 

C’(x) = 0.000009x^2 – 0.009x + 8

(Measured in dollars per unit). The daily fixed costs are found to be $120. 

a. Use this information to get a general cost function for producing widgets. 
b. Find the total cost of producing the first 500 widgets. 
c. If you sell the widgets for $25 each, how many will need to be sold before the 
Company begins making a profit. (Hint: The revenue function is R(x) = $25x;