hw8 - padilla(tp5647 – HW08 – Gilbert –(56650 1 This...

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Unformatted text preview: padilla (tp5647) – HW08 – Gilbert – (56650) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the integral I = integraldisplay 2 4 √ 16- x 2 dx . 1. I = 4 3 π 2. I = 2 3 π correct 3. I = π 4. I = 1 5. I = 2 3 6. I = 4 3 Explanation: Set x = 4 sin u ; then dx = 4 cos u du and 16- x 2 = 16(1- sin 2 u ) = 8 cos 2 u , while x = 0 = ⇒ u = 0 , x = 2 = ⇒ u = π 6 . In this case I = 4 integraldisplay π/ 6 cos u cos u du = 4 integraldisplay π/ 6 du . Consequently I = 2 3 π . 002 10.0 points Evaluate the integral I = integraldisplay 1 x 2 (2- x 2 ) 3 / 2 dx . 1. I = 2 parenleftBig √ 3 + π 3 parenrightBig 2. I = 2 parenleftBig √ 3- π 3 parenrightBig 3. I = √ 2- π 4 4. I = 2 parenleftBig √ 2 + π 3 parenrightBig 5. I = 1 + π 4 6. I = 1- π 4 correct Explanation: Let x = √ 2 sin θ . Then dx = √ 2 cos θ dθ , 2- x 2 = 2 cos 2 θ , while x = 0 = ⇒ θ = 0 , x = 1 = ⇒ θ = π 4 . In this case, I = integraldisplay π/ 4 2 √ 2 sin 2 θ cos θ 2 √ 2 cos 3 θ dθ = integraldisplay π/ 4 sin 2 θ cos 2 θ dθ = integraldisplay π/ 4 tan 2 θ dθ . Now tan 2 θ = sec 2 θ- 1 , d dθ tan θ = sec 2 θ , and so I = integraldisplay π/ 4 (sec 2 θ- 1) dθ = bracketleftBig tan θ- θ bracketrightBig π/ 4 . Consequently, I = 1- π 4 . padilla (tp5647) – HW08 – Gilbert – (56650) 2 003 10.0 points Evaluate the integral I = integraldisplay 1 1 (3 x 2 + 1) 3 / 2 dx . 1. I = 1 6 2. I = 1 3 3. I = 1 4. I = 1 2 correct 5. I = 1 4 Explanation: Set √ 3 x = tan u. Then √ 3 dx = sec 2 u du , while x = 0 = ⇒ u = 0 , x = 1 = ⇒ u = π 3 . On the other hand, (3 x 2 + 1) 3 / 2 = ( tan 2 u + 1 ) 3 / 2 = sec 3 u . Thus I = integraldisplay π/ 3 sec 2 u √ 3 sec 3 u du = 1 √ 3 integraldisplay π/ 3 cos u du = 1 √ 3 bracketleftBig sin u bracketrightBig π/ 3 . Consequently I = 1 2 . keywords: 004 10.0 points Evaluate the definite integral I = integraldisplay 2 √ 2 1 x 2 √ x 2- 1 dx . 1. I = √ 3- √ 2 2. I = √ 3 + √ 2 3. I = 1 2 ( √ 3 + √ 2 ) 4. I = 1 4 ( √ 3 + √ 2 ) 5. I = 1 4 ( √ 3- √ 2 ) 6. I = 1 2 ( √ 3- √ 2 ) correct Explanation: Set x = sec u . Then dx = sec u tan u du , x 2- 1 = tan 2 u , while x = √ 2 = ⇒ u = π 4 , x = 2 = ⇒ u = π 3 . In this case, I = integraldisplay π/ 3 π/ 4 sec u tan u sec 2 u tan u du = integraldisplay π/ 3 π/ 4 cos u du = bracketleftBig sin u bracketrightBig π/ 3 π/ 4 . Consequently, I = 1 2 ( √ 3- √ 2) . 005 10.0 points padilla (tp5647) – HW08 – Gilbert – (56650) 3 Which one of the following functions is an antiderivative of f when f ( x ) = 1 x 2- 8 x + 17 ?...
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hw8 - padilla(tp5647 – HW08 – Gilbert –(56650 1 This...

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