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Find the area of the region bounded by: a) y = sinx, the x-axis, x = 0 and x = π b) the axes and y = √ 9 - x unit2 unit2 c) How long will it take for the particle's velocity to increase o 45 m/s? b) Find the velocity of the particle after 3 seconds. a) State the initial velocity of the particle. The velocity of a particle travelling in a straight line is given by v(t) = 50 - 10e-0.5t, where t ≥ 0, t in seconds. m/s s m/s ( round off to 1 decimal place) ( round off to 1 decimal place) ∫ ∫ ∫ 1 4 1 ( x - e 1 - x dx = π/6 0 0 sin (3x) dx = √x 3 ) dx = Round all answer to 1 decimal place. The approximate value of ∫ Given the graph of f(x) -1 1 f(x) dx = ∫ (lnx)4 (lnx)4 ( lnx )3 4 4x x + C + C = (A) (B) (lnx)4 (lnx)4 4 4x + + x2 1 x2 1 + C + C (C) (D) ∫ (A) 6x2 - 2x 2x3 - x2 (C) ln | 2x3 -x| + C (B) ln | 6x2 -1| + C (D) 2 ln | x| + C 4x3 - 2x 2x3 - x dx = + C The mean value of a function f(x) from a to b is given by (A) (B) (D) (C) f(a) + f(b) f (a) + 2f ( ∫ ∫ b - a 2 a b a b f(x) dx f(x) dx 4 a+b 2 ) + f(b) The marginal profit for producing x dinner plates per week is given by P'(x) =15 - 0.03x dollars per plate. If no plate are made then a loss of $650 each week occur. Find the profit function P(x) , and hence find The maximum profit: $ displacement of the particle A particle is initially at the origin and moving to the right at 5 cm/s. It accelerates with time according to a(t) = 4 - 2t cm/s. For the first 6 seconds of motion, determine the total distance travelled (round to the second decimal place) cm cm Find the total area of the regions contained by f(x) = x3 + 2x2 - 3x and the x - axis (round to the second decimal place) unit2 Physically, integrating (A) area to the right of point a (B) displacement of a particle from a to b (C) area under the curve from a to b (D) total distance travel during the time b - a ∫ a b f(x) dx means finding the Find the area under f(x) from x =1 to x = 3 (round to the second decimal place) y = lnx Find the area of the shaded region (round to the second decimal place) Rotate y = sin x (0 ≤ x ≤ π)around the x - axis to getthe solid below . Findthe volume of the shape (round the the second decimal place). unit2 18x6 - 12x4 + 2x2 90x4 - 36x2 + 2 If F' = f and f(x) = 18x5 - 12x3 + 2x, which of the following could be F(x)? 3x6 - 4x4 + x2 + 5 3x6 - 3x4 + x2 + 1 The area of the shaded region is (decimal) Find the area of the shaded region: (round to 2nd decimal place) |