_________________________________________________________________________________
Calculus  II                                                          Test #1                                                  March 5, 2003

Name____________________                     R.  Hammack                                                Score ______
_________________________________________________________________________________
(1) Find the following integrals.

(a) ∫ (2x^3 + 1) dx =

(b) ∫x x^(1/2) dx =

(c) ∫ (4e^x + 1/x + sec(x) tan(x) ) dx =

(d) ∫1/(x^2 + 1) dx =

(e) ∫ (2x)/(x^2 + 1) dx =

(f) ∫ (3x + 2)^7dx =



(2) Find the following definite integrals.

(a)   ∫__ (-1)^1 (2x + 1) dx =

(b)    ∫__ (-π)^0cos(x) dx =

(c)  ∫_ln(3)^ln(4) e^x/(e^x + 4) dx =


(d)  ∫__0^1 (2x - 1)^4dx =




(3)  The expression   Underscript[lim , n∞](Underoverscript[∑ , k = 1, arg3] x_k ^* cos(x_k ^*)^(1/2) Δx )   represents a definite integral over the interval [3, 5].   Write the definite integral. (You do not need to find its value.)   

  



(4)  Find the average value of f(x) = x^2 on the interval [0, 3].






(5) Find the following integrals. You may find it easiest to consider the area under the graphs.

(a) ∫_ (-4)^0 (16 - x^2)^(1/2) dx = (π 4^2)/4 = 4π




(b) ∫_ (-2)^4 | x - 2 | dx =




(6) Find the derivative of the function F(x) =    ∫__3^x^4 (ln(t) + t  )^(1/2) dt




(7) A  train, moving with constant acceleration, travels 25 miles in half an hour.  At the beginning of the half-hour period, it has a velocity of 10 miles per hour.  What is its velocity at the end of the half-hour period?





(8) Suppose that    ∫_3^2f(x) dx = 3  and  ∫_2^6f(x) dx = 9.

(a)    ∫_2^35f(x) dx =


(b)    ∫_3^6f(x) dx =