Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration P1^ 6 16  (i) Sketch the curve with equation y = x^3 − 6 x^2 + 11 x − 6 for 0  x  4. (ii) Shade t ...

P1^ 6 Areas below the x axis 20  A curve is such that d d y x x =^163 ,^ and^ (1,^ 4)^ is^ a^ point^ on^ the^ curve. (i) Find th ...

Integration P1^ 6 EXAMPLE 6.11 Find the area of the region bounded by the curve with equation y x =−^223 , the lines x = 2 ...

P1^ 6 Areas (^) below (^) the (^) x axis EXAMPLE 6.12 Find the area between the curve and the x axis for the function y = x^2 + ...

Integration 196 P1^ 6 EXERCISE 6C  1  Sketch each of these curves and find the area between the curve and the x axis between the ...

P1^ 6 The area between two curves The area between two curves EXAMPLE 6.13  Find the area enclosed by the line y = x ...

Integration P1^ 6 A = B − C = (^) ∫ 3 0 (x^ +^ 1)^ dx^ −^ ∫ 3 0 (x (^2) − 2 x + 1) dx (^) =+     +    =+()  x^2 ...

P1^ 6 Exercise (^) 6D 2  (i) Sketch the curves with equations y = x^2 + 3 and y = 5 − x^2 on the same axes, and shade ...

Integration P1^ 6 12  The diagram shows the curve with equation y = x^2 (3 − 2 x − x^2 ). P and Q are points on the curve with ...

P1^ 6 Exercise (^) 6D 14  The diagram shows the curve y = (x − 2)^2 and the line y + 2 x = 7, which intersect at points A and B ...

Integration P1^ 6 The area between a curve and the y axis So far you have calculated areas between curves and the x ax ...

P1^ 6 The reverse chain rule EXAMPLE 6.15 Find the area between the curve y = x and the y axis between y = 0 and y = 3. SOL ...

Integration P1^ 6 (ii) Use your answers to part (i) to find these. (a) (^) ∫ 42 ()xx−^3 d (b) (^) ∫()xx− 23 d (c) (^) ∫ 72 ()xx+ ...

P1^ 6 Exercise (^) 6F EXERCISE 6F 1   Evaluate the following indefinite integrals. (i) (^) ∫()xx+ 54 d (ii) (^) ∫()xx+ 78 d (iii ...

Integration 206 P1^ 6 5  Find the area of the shaded region for each of the following graphs. (i) (ii) 6  The equation of a cur ...

P1^ 6 Improper (^) integrals 207 At first sight, (^1112) x 1 x x ∞ ∞ ∫ =−     d doesn’t look like a particularly daunting ...

Integration P1^ 6 You can see that the expression is undefined at x = 0, so you need to find the integral from a to 9 and th ...

Finding (^) volumes (^) by (^) integration 209 P1^ 6 ●?^1 Describe^ the^ solid^ of^ revolution^ obtained^ by^ a^ rotation^ throu ...

Integration P1^ 6 ! Since the integration is ‘with respect to x’, indicated by the dx and the fact that the limits a and b are ...

Finding (^) volumes (^) by (^) integration 211 P1^ 6 Solids formed by rotation about the y axis When a region is rotated about t ...