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Unformatted text preview: Techniques of Integration 1. Evaluate the following definite integrals. (a) Z 2 x 2 e x 3 dx (b) Z 1 x 2 e 3 x dx 2. Evaluate the following integrals, showing all of your work. (a) Z x 3 √ 1 x 2 dx (c) Z x 2 (ln x ) 2 dx (b) Z sin 3 x cos 3 x dx (d) Z sin 4 x dx 3. Evaluate the following integrals, showing all of your work. (a) Z 12 4 x √ 1 + 2 x dx (c) Z √ t e √ t dt (b) Z ln x x 1 / 3 dx (d) Z π/ 2 π/ 2 x cos 2 x dx 4. Find each of the following, showing all of your work. (a) Z tan x dx Hint: tan x = sin x cos x (c) Z (sin 2 x cos 2 x + sec 2 x ) dx (b) Z x √ 4 + x dx (d) Z ln x x 2 dx 5. Evaluate the following integrals. Use whatever method you like, but be sure to show all work. (a) Z dx 3 x (1 x 20 ) (b) Z t ( t 10 + 1) t 4 + 5 dt 6. Evaluate the following integrals. Use whatever method you like, but be sure to show all work. (a) Z dx 2 x 2 6 x (b) Z dx x 3 √ 9 x 4 7. See problem 10(c) in the “Volumes” section below to practice another substitutionstyle inte gral. Approximate Integration 1. Consider the function f , whose formula and derivatives are given below: f ( x ) = 1 1 + x 2 f ( x ) = 2 x (1 + x 2 ) 2 f 00 ( x ) = 2 + 6 x 2 (1 + x 2 ) 3 f 000 ( x ) = 24 x ( x 2 1) (1 + x 2 ) 4 (a) Let J = Z 1 f ( x ) dx . Write an expression involving only numbers that estimates the value of J using the Trapezoidal Rule with n = 6 subintervals. (You do not have to simplify this expression.) (b) Calculate the value of the “error bound” associated with the approximation above, and explain its significance in a complete sentence. Be as mathematically precise as you can in your reasoning; however, you don’t have to simplify all your arithmetic. (You may find the derivatives of f provided above to be helpful.) (c) Now suppose you want to make a Trapezoidal Rule approximation of our J = Z 1 f ( x ) dx that is sure to be within 10 8 of the true value. How many subintervals would you use? Explain completely; simplify your answer as much as possible. (d) Now consider an arbitrary function g ( x ). How does the graph of g affect whether an approximation by the Trapezoidal Rule is an overestimate or an underestimate? Ex plain why this is so. (It might help to draw a picture, but this alone is not sufficient justification.) 2. Let h be the function graphed below. 1 2 1 2 3 h Four students (I, II, III, and IV) approximated the area under the graph of h from x = 0 to x = 2. They all used the same number of subin tervals, but they each used a different method among the ones listed below. Here are their re sults: I: 5 . 4386 II: 5 . 70486 III: 5 . 73442 IV: 5 . 97112 (a) Which result corresponds to which method? Explain. Method I, II, III, or IV? Brief reason Left Endpoint Rule Right Endpoint Rule Midpoint Rule Trapezoidal Rule (b) Write an expression, involving h evaluated at specific numbers, that represents the Simp son’s Rule approximation to the area R 2 h ( x ) dx using n = 8 subintervals....
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 '07
 Butscher,A
 Calculus, Integrals, dx, sample slice

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