= = – A = square units Find the area between the curve 4x and the from: y = x2 + x-axis = 2 to = 0 x x – = 2 to = 2 x x – Solution: Let A1 = = = 0 –[ = 0 [ ] – = – …..the minus sign tells us the area is below the x-axis Thus, A1 = For the interval [– 2 ; 2], we observe that the required area comprises of two parts – one part below the x-axis and the other part above the x-axis. Calculate A2 , as follows: A2 = = (The reader should verify the accuracy) So, the area over the interval [–2 ; 2] is obtained by adding A1 and A2, i.e: A = A1 + A2 = + = 16 square With your partner, attempt the following. The given sketch is a graphical representation of the function defined by: = 4x2 + 3x. Determine the area f(x) x3 – between the curve and the from = 0 to = 3. x-axis x x Find the area enclosed between the curve y = 2x and the (A challenge) x2 – –3 straight-line y = + 1. x Activity 9 1. Use the properties of definite integrals which you have studied thus far, to evaluate the following definite integrals. a) b) c) d) 2. Calculate the areas of the following: a) b) c) d)
= = – A = square units Find the area between the curve y = x2 + 4x and the x-axis from: x = – 2 to x = 0 x = – 2 to x = 2 Solution: Let A1 = = = 0 –[ = 0 – [ ] = – …..the minus sign tells us the area is below the x-axis Thus, A1 = For the interval [– 2 ; 2], we observe that the required area comprises of two parts – one part below the x-axis and the other part above the x-axis. Calculate A2 , as follows: A2 = = (The reader should verify the accuracy) So, the area over the interval [–2 ; 2] is obtained by adding A1 and A2, i.e: A = A1 + A2 = + = 16 square With your partner, attempt the following. The given sketch is a graphical representation of the function defined by: f(x) = x3 – 4x2 + 3x. Determine the area between the curve and the x-axis from x = 0 to x = 3. (A challenge) Find the area enclosed between the curve y = x2 – 2x –3 and the straight-line y = x + 1. Activity 9 1. Use the properties of definite integrals which you have studied thus far, to evaluate the following definite integrals. a) b) c) d) 2. Calculate the areas of the following: a) b) c) d)
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