2010-h2 maths-integration and its applications (tut 3) - int applications solutions)

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National Junior College Mathematics Department 2010 2010 / SH1 / H2 Maths / Integration and its Applications (Tutorial 3) – Applications 1 National Junior College (Suggested Solutions) 2010 H2 Mathematics (Senior High 1) Integration and its Applications (Tutorial 3) – Applications Basic Mastery Questions 1(a) [ ] ( ) ( ) sin d cos cos cos 0 x x x π π π π π π − − = − = − − − − = ∫ 1(b) Area of region = [ ] ( ) 0 0 sin d cos cos cos 0 2 x x x π π π = − = − − − = ∫ 1(c) Area of region = [ ] ( ) 0 0 sin d cos cos 0 cos 2 2 x x x π π π − − = − = − − − = − = ∫ It is the same answer as in 1(b). 1(d) Area of region required = 4 The answer in 1(a) is zero since the definite integral is negative when curve is below the x-axis and is of equal magnitude as that when curve is above the x-axis. Hence, the two answe rs cancel out. When finding the area, we are intereste d in only the magnitude. 2 (i) Volume obtained when rotated about the x-axis ( ) ( ) ( ) ( ) 2 2 2 2 2 4 2 0 0 2 5 3 0 5 2 1 d 50 2 1 d 28 2 32 544 50 50 2 5 3 5 3 15 x x x x x x x x π π π π π π π π π = − + = − + + = − + + = − + + = ∫ ∫ (ii) Volume obtained when rotated about the y-axis ( ) ( ) 5 2 5 5 2 1 1 1 2 2 d 1 d 2 5 1 5 1 8 2 2 y x y y y y π π π π π = = − = − = − − − = ∫ ∫ R 2 1 y x = + 5 y =

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