SPM Add Math Form 5 Chapter 3 Integration

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1 SPM Add Math Form Chapter Integration INDEFINITE INTEGRAL CHAPTER : INTEGRATION Integration as the reverse process of differentiation ) y if dy = x. Given that d Integral of ax n x + c = x, where c is a constant, find x. ) Integrate the following with respect to x. x x (c) -x (d) -x (e) 7 (f) -0 ) Integrate the following with respect to x. x x (c) 8 x (d) x ) If dy =, find y. 7x Integrals of algebraic expressions (e) x 7 (f) ) the following indefinite integrals. x + (c) x x (e) x x + x x (d) x (f) ) x + x. ing the constant of integration [SPM 998 P clone] 7) If dy = x x and y= when x=, find the constant of integration, c. Equation of curves from functions of gradients x + 8) the equation of a curve whose gradient is x and passes through the point (,). 9) the equation of the curve which passes through the point (,-) and for which dy = x x. [SPM 998 P clone] 0) Given that dy = x x + and y= when x=, find y in terms of x. [SPM 00 P clone] ) A curve has a gradient function kx +x where k is a constant. The tangent to the curve at the point (, ) is parallel to the straight line y=x-7. the value of k, the equation of the curve. x [SPM 00 P clone] Copyright Page

2 SPM Add Math Form Chapter Integration Integration of (ax+b) n ) Integrate the following with respect to x. (x+) (x ) (c) ( x) (d) (x ) ) Given that the gradient of a curve is the equation of the curve. 8 (x ) and that the curve passes through the point (,-). ) Given that (x+) = k(x + )n + c, find the values of k and n. [SPM 00 P clone] DEFINITE INTEGRAL Definite integrals of algebraic expressions ) Evaluate x 0 (x 8 x ) (c) x + x (d) x (x + ) ) Evaluate (x ) (x ) 7) Given that d f x = x, find the value of f f. 8) Given that y = x dy and g x. x = g x such that g(x) is a function of x. the value of 9) Show that d x x = ( x). Hence or otherwise, evaluate ( x ). 0) Given that f x = and f x =, find f x f x (c) f x (d) f x + x ) Given that x =, evaluate x x, the value of k for which x + kx =. ) Evaluate x + 8x. [SPM 00 P clone] ) Given that d x x = f(x), calculate the value of f x + x 0. [SPM 00 P clone] ) Given that x =, k where k>0, find the value of k. [SPM 00 P clone] Copyright Page

3 SPM Add Math Form Chapter Integration ) Given that f x = and f x + kx =, find the value of k. [SPM 00 P clone] ) Given that x = 8, find the value of x, the value of p if px x =. [SPM 00 P clone] Areas under curves using formula 7) For each of the following diagrams, find the area of the shaded region. 8) The diagram shows part of the curve y = x. the area of the shaded region. 9) The diagram shows part of the curve y = and of the line y=x intersecting at A. the coordinates of A, the area of the shaded region. x 0) For each of the following diagrams, find the area of the shaded region. ) The diagram shows the shaded region, enclosed by the curve x = y(y ) and the y- axis. Calculate the area of the shaded region. Copyright Page

4 SPM Add Math Form Chapter Integration ) The diagram shows part of the curve y = x And of the line y=-x intersecting at point P. the coordinates of P, the area of the shaded region. ) The diagram shows part of the curve y = x x + 8 and part of the line y=x. the coordinates of the points P and Q, the area of the shaded region. ) The diagram shows part of the curve y = 9x and of the line x=, intersecting at the points P and Q. the coordinates of P and Q, the area of the shaded region. ) The diagram shows part of the curve y = f(x) which touches the x-axis at A and cuts the y-axis at B. The tangent to the curve at the point B and the x-axis are parallel. Given that f x = x x +, find the coordinates of A, f x, (c) the coordinates of B, (d) the area of the shaded region. [SPM 999 P clone] ) The diagram shows a straight line y = f(x) which intersects the curve y = (x + ) at the point (-,9). Given that f x = 7, find the area of the shaded region. [SPM 00 P clone] 0 7) The diagram shows part of the curve y = x and the straight line x=k. If the area of the shaded region is 8 unit, find the value of k. [SPM 00 P clone] Copyright Page

5 SPM Add Math Form Chapter Integration 8) The diagram shows the curve y = f(x) cutting the x-axis at x=a and x=b. Given that the area of the shaded region is 8 unit, find the value of a b f x. [SPM 00 P clone] Volumes of revolutions using formula 9) For each of the following shaded regions, find the volume of the solid generated by rotating 0 about the x-axis. 0) For each of the following shaded regions, find the volume of the solid generated by rotating 0 about the y-axis. ) The diagram shows part of the curve y = x and of the line y=x+ intersecting at P. the coordinates of P, the volume of solid formed by rotating the shaded region 0 about the x-axis. ) The diagram shows part of the two curves y = x and y = 8 x intersecting at A. the coordinates of A, the volume generated when the shaded region is rotated through 0 about the y-axis. Copyright Page

6 SPM Add Math Form Chapter Integration ) The diagram shows the curve y = x and the line x y =. the volume generated when the shaded region is rotated 0 about the y-axis. [SPM 998 P clone] ) The diagram shows part of the curve y = x +. The shaded region is bounded by the curve, the y-axis and the line y=k. If the volume of revolution obtained by rotating the shaded region through 0 about the y-axis is π unit, evaluate k. [SPM 00 P clone] Copyright Page