lab9_f11 - Math 116 - Lab 9 - Fall 2011. Integration by...

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Unformatted text preview: Math 116 - Lab 9 - Fall 2011. Integration by Substitution, Average Value of a Function, Curve Length (Sections 6.1, 7.7 and 7.1 of L/G). You are to provide full solutions to the following problems. You are allowed to collaborate with your classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions is not encouraged, nor is it allowed or ethical. First name: Last name: Student number: 1 Math 116 - Lab 9 - Fall 2011. Student number: 1. Compute the following integrals by substitution. a) b) c) d) e) f) g) h) i) j) k) l) sin(3x) dx exp(t + 3) dt (3t3 + 7)4 t2 dt t dt 2+5 3t 3x2 dx (Hint: recall the derivative of arctan(x).) 1 + x6 1 dt (Hint: recall the derivative of arctan(x) and complete the square.) t2 + 2t + 5 t exp(t2 + 1) dt (Hint: recall the derivative of arctan(x).) 1 + exp(2t2 + 2) tan(t) dt cos(t) cos(t) dt sin(t) t dt 2 (t2 + 1) cos sin2 (t) cos(t) dt sin5 (t) cos10 (t) dt 2 2t exp(t2 ) dt m) 0 2 n) 1 1 dt (t + 1)2 −1 o) 3 cosh(2x + 1) dx 0 1 √ 2 p) 0 6t √ dt (Hint: recall the derivative of arcsin(x).) 1 − t4 2. Compute 2 sin(x) cos(x) dx in three different ways: use the substitution u = sin(x), use the substitution u = cos(x), and use the trigonometric identity sin(2x) = 2 sin(x) cos(x). Explain why the three results you obtain are essentially the same result. 3. a) Determine the average value of f (x) = x + 7 over [1, 2] in two different ways (by integration and graphically). b) Determine the average value of f (t) = ln(t)/t over [1, 3]. 4. a) Determine the curve length of f (x) = x3/2 between x = 0 and x = 1. b) Determine the curve length of f (x) = cosh(x) between x = − ln(2) and x = ln(2). (Hint: the integrand simplifies considerably. . . ) 2 ...
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This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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lab9_f11 - Math 116 - Lab 9 - Fall 2011. Integration by...

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