sol_test1 - 1 Solutions to Test 1 MAT1332 A Fall 2009 2 1[4...

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1 Solutions to Test 1 - MAT1332 A- Fall 2009 1. [4 marks] Compute the definite integral 2 Z 1 2 x 2 + x 2 2 dx . Solution : 2 Z 1 2 x 2 + x 2 2 dx = - 2 x + x 3 6 2 1 = - 2 2 + 8 6 + 2 - 1 6 = 13 6 2. [4 marks] Use substitution to find the indefinite integral Z x 3 (3 x 4 + 50) 31 dx . Solution : Let u = 3 x 4 + 50. Then du = 12 x 3 dx , that is, x 3 dx = 1 12 du . Thus: Z x 3 (3 x 4 + 50) 31 dx = 1 12 Z u 31 du = 1 12 · u 32 32 + C = (3 x 4 + 50) 32 384 + C, where C is a constant. 3. [5 marks] Use indefinite integral to solve the following differential equa- tion: dV dt = 4 t 3 + 3 t 2 + 2 t + 1 with V (1) = 6 . Solution : V ( t ) = Z (4 t 3 + 3 t 2 + 2 t + 1) dt = 4 t 4 4 + 3 t 3 3 + 2 t 2 2 + t + C = t 4 + t 3 + t 2 + t + C where C is a constant. Since V (1) = 6, we obtain that V (1) = 1 4 + 1 3 + 1 2 + 1 + C = 6 = 4 + C = 6 , and so C = 2. Hence V ( t ) = t 4 + t 3 + t 2 + t + 2. 4. [6 marks] Find the area of the region that is enclosed between the curves y = x 2 and y = x + 6. ( Hint: first find the points of intersection of the two curves ) Solution : We first need to find the points of intersection of the two curves.
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