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Calculus Spring Exam Review Formulas n(n +1) åi = 2 i =1 n n(n +1)(2n +1) åi = 6 i =1 n å c = cn i =1 n 2 n 2 (n +1)2 åi = 4 i =1 n 3 d u du éë a ùû = ( ln a ) au dx dx d 1 du log a u ] = [ dx ( ln a ) u dx ò tanu du = -ln cosu + C ò cot u du = ln sinu + C ò secu du = ln secu + tanu + C ò cscu du = -ln cscu + cot u + C òa x 1 ö x dx = æç a +C è ln a ÷ø Chapter 4 1. Use the properties of sigma notation and the above summation formulas to evaluate 10 å (i the sum: 2 + 3i - 2). i=1 2. (SKIP) Write the definite integral that represents the shaded region. 5 3. Draw a sketch representing ò (-2x + 6)dx. -1 4. If 6 12 12 -1 6 6 ò f (x)dx = 5 , ò f (x)dx = -3, and ò f (x)dx = 6, find each using properties of the definite integral. 12 a) Find ò 12 f (x)dx b) Find -1 ò [ f (x) + 3]dx 6 5. Find the average value of f ( x) 3x 2 on the interval [0, 2]. 2 6. Find f(x) if f " ( x) x 2 , f ' (0) 3 , f (0) 1 7. Use a(t ) 32 feet per second squared as the acceleration due to gravity. A ball is thrown vertically upward from the ground with an initial velocity of 56 feet per second. For how many seconds will the ball be going upward? 3 8. Use the Trapezoidal Rule with n = 4 to approximate 1 ò ( x - 1) dx. 2 2 9. Evaluate the following integrals by hand (not with a calculator). Give an EXACT value (decimal approximations are not acceptable) for definite integrals. a. b. ò 3csc c. x4 x3 x 2 dx d. ò (3x e. sec 3 q tanq ò 1+ tan2 q dq f. ò 2 cos ( 3x )dx g. ò 3x x 2 dx 5 2 xdx + 1)2 dx 2 4x 2 -1dx h. (SKIP) ò 3x 4x - 1dx i. ò sin(3x + 4)dx 2 j. 1 x 2 dx 1 p k. 3 ò csc x cot xdx p 4 x2 + 5 ò x 2 dx -1 1 l. m. ò x(x n. ò sin 3 2 - 1)4 dx 3x cos 3xdx 1 o. ò x dx -1 p. SKIP ò x x + 1dx Chapter 5 10. Find y. a. f (x) = ln b. y = x e x 2 4x + 1 (x 3 + 5) 3 x c. f ( x) e2x 2x d. f ( x) (3 sin x 5) 4 e. f (x) = x ln x f. f (x) = 4x 2 e3x g. j(x) = cos3(2x) h. ln xy x y i. xy 3 - 3x = 2y - 5 j. y = 3xx3 11. Evaluate. 4e a. 1 ò x dx e e+1 b. 1 ò x - 1dx 2 2x + 1 dx x +1 c. ò d. ò x cot x dx e. 3x 2 + 3x + 3 ò x 2 +1 dx 2 x f. ò e g. ò ln x dx x h. cos 3 x - sin 2 x ò cos2 x dx i. ò tan x 5 dx x sec 2 xdx 12. Find the general solution to the first order differential equation: (4-x)dy + 2ydx = 0. 13. A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many will there be 5 hours from the initial time given? Chapter 7 14. Determine the area of the region bounded by the graphs of y = -x2 + 2x + 3 and y = 3. 15. Find the area of the two regions bounded by y = 3 – x2, y = 3 – x, and x = -2. 16. Find the volume of the solid formed by revolving the region bounded by the graphs of y = x3, x = 2, and y = 1 about the y-axis. 17. What integral represents the volume of the solid formed by revolving the region bounded by the graphs of y = x3, y = 1, and x = 2 about the line y = 10? 18. Find the volume of the solid formed by revolving the region bounded by the graphs 1 of y = (x - 2)2 and y = 2 about the y-axis. 2 19. Find the volume of the solid formed by revolving the shaded region enclosed by y = x , y = 4x , and x = 4 about the line y = 4. 20. Let R be the region bounded by f(x) = x2 and y = 2. a. Sketch the region R. b. Find the area of R. c. Revolve R around the x-axis and find the volume of the solid generated. Cross sections