Akos_M101_10prac1

# Akos_M101_10prac1 - x for the following quantities(a The...

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MATH 101 - MIDTERM - PRACTICE QUESTIONS. Problem I. (12 pts) Short answer questions. Here you only need to give the correct answer no detailed explanations/ calculations are needed to support your answer. (a) (2 pts) Evaluate Z π - π sin 3 x cos 2 x dx (b) (2 pts) Express the limit as a deﬁnite integral lim n →∞ 1 n X i =1 n 2 n 2 + i 2 (c) (3 pts) Let R be the region between the curve y = sin 2 x and the x-axis, for 0 x π 2 . Find the area of the region. Simplify your answer completely! (d) (3 pts) If R 2 0 f ( x ) dx = 1 then ﬁnd Z π/ 4 0 f (2 tan θ ) sec 2 θ dθ (e) (2 pts) Write down the n-th right Riemann approximating sum for the integral Z 2 1 ( x + 1 x ) dx Full Answer questions. Here you have to give detailed explanations and/or calculations to support your answer. Problem II. (8 pts) Evaluate the following indeﬁnite integrals ( a ) (3 pts ) Z x 3 p x 2 + 1 dx 1

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2 MATH 101 - MIDTERM - PRACTICE QUESTIONS. ( c ) (5 pts ) Z π/ 4 0 tan 3 ( x ) dx Hint: The identity sec 2 x = 1 + tan 2 x might be useful. Problem III. (6 pts) Let R be the region bounded by the curves y = x 2 and y = 2 x - x 2 . Set up an integration in the variable
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Unformatted text preview: x for the following quantities: (a) The volume obtained by revolving R about the x-axis (b) The volume obtained by revolving R about the y-axis (c) The volume obtained by revolving R about the line x =-2 Problem IV. (4 pts) A solid has circular base of radius 1. Find the volume of the solid if the cross-sections perpendicular to the base are isosceles right triangles with hypotenuse in the base Problem V. (4 pts) Prove the following inequality. Explain all steps in your proof: Z π/ 4 sin( x 2 ) dx ≤ 1-1 √ 2 Hint: Compare the integrand to a simpler function. Problem VI. (6 pts) Find the area A of the region under the graph y = 2 x-x 2 above the interval [0,2], by completing the following steps. a) (2 pts) Write a formula for the n-th approximating Riemann sum to the area. b) (4 pts) Evaluate and simplify the sums you obtained and ﬁnd the limit as n → ∞ ....
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Akos_M101_10prac1 - x for the following quantities(a The...

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