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-2 2 -2 Figure 8.6: Negative integrands and their relationship to area (continued). For Example 8.11, the region between the parabola y = 1 -r, the r-axis and the vertical lines x = -2 and r = 2 Learning outcomes At the end of this unit, you should be able to: explain the relationship between differentiation and indefinite integration; find simple indefinite integrals by using standard integrals and the rules of integration; explain the relationship between a definite integral and an area; find areas using definite integration. Exercises Exercise 8.1 Find the following indefinite integrals and use differentiation to verify your answer. i. -17 dr; vi. 5edx; ii. 27x di; vii. (322 - 5x + 7) dr; ifi. 2x' dr; viii. (3210) + 825 + 4e') dr; iv . 5Vidr; ix. (3V/73 - 2x-1) dr; v. -dr; x. 1( 3+2) 144 8. Calculus IV - Integration Exercise 8.2 Differentiate the function F(x) = x In(x) - r. (You will need to use the product rule!) Hence find In(2) dz, by thinking of the indefinite integral in terms of antiderivatives. Exercise 8.3 Use the 'adding powers' power law and the constant multiple rule to show that with do = efth + c , where c is an arbitrary constant. Exercise 8.4 Find the indefinite integral (2x - 1)2 dx, by multiplying out the brackets and integrating term-by-term. Exercise 8.5 Evaluate the following definite integrals. 1. ( 2dr; v . [3Vidz; ii. vi. ( 423 + 3 ) dr; ini. vii. iv . [ ze' dr; vili. " sin(2 ) da. Exercise 8.6 Find the areas between the following curves, the r-axis and the vertical lines r = 1 and r = 3. (You may find it useful to sketch the curve in each case.) i. y= x - x -2, ii. y =x -3