Summer Term I Kostadinov. MA124 Calculus II Boston University. Evaluate the definite integrals. sin(ln(x)) x

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Bartholomew Barker
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1 student: Exam I Problem : Evaluate the indefinite integrals 2e x + cos(x) dx 8x x dx Problem 2: Evaluate the definite integrals 4 3 x + x dx π/2 π/6 sin(x) dx

2 Problem 3: Evaluate the definite integrals e π/2 sin(ln(x)) x dx 0 xe 3x2 dx Problem 4: Evaluate the indefinite integrals x 2 cos(2x) dx f(x)f (x) dx

3 Problem 5: Evaluate the definite integral 2 5x 2x 2 + 3x 2 dx Problem 6: 5x x 2 x 2 4 dx Evaluate the indefinite integral

4 Problem 7: Evaluate the definite integral 2/π /π e x sin( x ) x 2 dx Problem 8: Evaluate the indefinite integral 2x 3 dx e x2

5 Problem 9: Apply the interpretation of a integral of a function with positive values as the area under the graph of the function to express the area under the graph of f(x) in problem 2 bounded by the vertical lines x = 6 and x = 0. Then use Simpson s Rule to approximate this area. Problem 0: Apply the definition of improper integral to decide whether the following integrals are convergent or divergent: 3 x 2 dx 2 (x ) 2 dx

6 Problem : Find the derivatives of the functions T (x) and T 2 (x). State the reason of your answers. a) T (x) = 0 2t dt x b) T 2 (x) = e 3x ln(t2 )dt 2x Problem 2: shown. Let g(x) = x f(t)dt, where f is the function whose graph is Evaluate the function values f(0) = g (2) = g() = g(4) = Where g(x) has a MINIMAL value in the interval [,9]? (Circle the right answer) x = x = 2 x = 3 x = 4 x = 5 x = 6 x = 8 x = 9 On what interval g decreases?

7 Problem 3: Sketch the region bounded by the curves: y = x 2 and y = 3x 2 5x Label the curves and determine any points of intersection. Then find the area of this region. Problem 4: Sketch the region bounded by the curves y = e x, y = 0, x = 0, x =. Label the curves and determine any points of intersection. Find the volume of the solid obtained by rotating this region about the x-axis.

8 Problem 5: equations: Find the exact length of the curve given by the parametric x = e t cos(t), y = e t sin(t), 0 t π Problem 6: Let f(x) = for x in the interval [,e]. Find the average value x of f on the given interval. Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f. Find a number c, such that the average value of f is f(c).

9 Problem 7: A concert hall has been charging $50 per person and selling about 400 tickets on a typical weeknight. After surveying there customers, the company estimates, that for every $5 they lower the price, the number of concertgoers will increase by 40 per night. Find the demand function and calculate the consumer surplus when tickets are priced at $30. Problem 8: A particle is moved along the x-axis by a force that measures 3 N at a point x meters from the origin. Find the work done in moving (2+x) 3 the particle from the origin to a distance meter.

10 Problem 9: State both parts of the Fundamental Theorem of Calculus. Problem 20: Suppose that Sue runs faster than Kathy throughout a 500- meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?

11 : 2: 3: 4: 5: 6: 7: 8: 9: 0: : 2: REFERENCE TABLE a2 + u 2 du = u a2 + u a2 ln(u + a u 2 ) + C du u = ln(u + a 2 + u 2 ) + C 2 +a 2 du u 2 = a2 +u 2 + C u 2 +a 2 a 2 u a 2 +u 2 du = a 2 +u 2 + ln(u + a u 2 u 2 + u 2 ) + C u 2 du u = u u2 a 2 a a2 ln u + u 2 2 a 2 + C u 2 du u a 2 = 2 5 (8a4 + 3u 2 + 4a 2 u) u a 2 + C du u 2 = u 2 a 2 u 2 a 2 a 2 u + C sin(au) cos(bu) du = cos((a b)u) 2(a b) cos((a+b)u) 2(a+b) + C u 2 du a = u a2 + u 2 +u a2 ln(u + a u 2 ) + C e au cos(bu) du = eau (a cos(bu) + b sin(bu)) + C a 2 +b 2 ue au du = (au )e au + C a 2 du u = a+ ln( a 2 +u 2 ) + C a 2 +u 2 a u Values of trigonometric functions, you may find useful: sin(0) = 0 sin π 6 = 2 sin π 4 = 2 2 sin π 3 = 3 2 Various formulas you may find useful: sin π 2 = sin π = 0 cos(x) = sin( π 2 x) cos( x) = cos(x) cos(2x) = 2 sin2 (x) ln(x) ln(y) = ln( x y ) a ln(b) = ln(ba ) ln(e) =

Practice Exam I. Summer Term I Kostadinov. MA124 Calculus II Boston University

Practice Exam I. Summer Term I Kostadinov. MA124 Calculus II Boston University student: Practice Exam I Problem 1: Find the derivative of the functions T 1 (x), T 2 (x), T 3 (x). State the reason of your answers. a) T 1 (x) = x 2t dt 2 b) T 2 (x) = e x ln(t2 )dt c) T 3 (x) = x 2