232sampleexam1

232sampleexam1 - Z x 2 cos 3 xdx 6(8 pts Find the area...

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MA 23200 NAME: Exam 1 PUID: INSTRUCTIONS No books or notes are allowed. You may use a one-line scientiﬁc calculator. No other electronic device is allowed. Be sure to turn oﬀ your cellphone. Show all your work in the space provided. Little or no credit may be given for an answer with insuﬃcient or inconsistent work, even if the answer happens to be correct. Write answers in the boxes provided. All answers are expected to be simpliﬁed ( 2 4 1 2 , 2 x + x 3 x , e ln2 2, etc). Question Possible Score 1 8 2 8 3 10 4 10 5 10 6 8 7 8 8 10 9 10 10 10 11 8 Total 100 The Trapezoid Rule for estimating the integral R b a f ( x ) dx with n trapezoids is given by T n = 1 2 Δ x [ f ( x 0 ) + 2 f ( x 1 ) + ··· + 2 f ( x n - 1 ) + f ( x n )] where Δ x = b - a n , x 0 = a , x 1 = a + Δ x , x 2 = a + 2Δ x , ..., x n = a + n Δ x = b . 1

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For Problems 1 - 5, evaluate the given integrals. On the problems 1 through 3, use the substitution method. On problems 4 and 5, use integration by parts. 1.) (8 pts) Z e 3 x 1 + 2 e 3 x dx 2.) (8 pts) Z 1 0 (5 x - 2) 4 dx 2
3.) (10 pts) Z 2 1 x 2 (1 + x 3 ) 2 dx 4.) (10 pts) Z 5 1 ln t t 2 dt 3

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5.) (10 pts)

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Unformatted text preview: Z x 2 cos 3 xdx. 6.) (8 pts) Find the area under the graph of the function f ( x ) over the interval [-1 , 4] where f ( x ) = ± 2-x, if x < 1 x 2 , if x ≥ 1 . 4 7.) (8 pts) Approximate the integral Z 1-1 1 1 + x 2 dx using the Trapezoid Rule with n = 4. (See page 1.) T 4 = 8.) (10 pts) Determine whether the integral Z ∞ xe-4 x dx is convergent or divergent. If it converges, calculate its value. 5 9.) (10 pts) Find the volume generated by revolving about the x-axis the region under y = 1 x + x from x = 1 to x = 5. V= 10.) (10 pts) Find the volume of the solid that, for 0 ≤ x ≤ 2, has cross-sections that are triangles with base 2 x 2 and height 3 x + 1. V = 6 11.) (8 pts) Evaluate the following integral by substitution: Z x √ x + 1 dx Hint: Use integration by parts. 7...
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232sampleexam1 - Z x 2 cos 3 xdx 6(8 pts Find the area...

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