Definite integrals and area under a curve: OCR A‑Level Maths questions

OCR A-Level Maths past-paper questions on 1.08.2 Definite integrals and area under a curve, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 16 questions on Definite integrals and area under a curve, as a list

June 2025, Paper 3, Q5 (14 marks): (a) Use the substitution u cos x to show that = 1 cos x x + y d ln ( 1 cos x ) c . [4] = - + sin x Fig. 1 y x a O 1 cos x + Fig. 1 shows part of the curve y . =
June 2025, AS Paper 1, Q2 (3 marks): y 2 2 It is given that ( ax 2 x 4 )d x 45 , where a is a constant. - + = 0 Determine the value of a. [3]
June 2025, AS Paper 2, Q7 (9 marks): y (1, 0) a x O (3, –16) The diagram shows the curve with equation y f ( ) x where ( ) f x is a quadratic function. The = curve intersects the x-axis at the poin
June 2024, AS Paper 2, Q6 (7 marks): y x O 2 P 5 4 3 The diagram shows the curve with equation y x ax bx , where a and b are integers. The = + + curve has a minimum at the point P where x 2 . = The
June 2023, Paper 2, Q3 (3 marks): In this question you must show detailed reasoning. 1 Find the exact area of the region enclosed by the curve y + , the two axes and the = x 2 line x 2 5 . . [3]
June 2023, AS Paper 1, Q5 (6 marks): In this question you must show detailed reasoning. 3 The diagram shows part of the graph of y x 4 x . = - y x Determine the total area enclosed by the curve and
June 2023, AS Paper 2, Q8 (7 marks): In this question you must show detailed reasoning. a 4 c 4 m y x Given that 3 d 7 , find the value of a. [7] + = x
June 2022, Paper 2, Q3 (10 marks): 3 (a) Amaya and Ben integrated ( 1 )x , with respect to x, using different methods, as follows. + 3 x ( 1 ) + y 2 1 2 1 3 Amaya: ( 1 x ) d x c x x x c + = + = +
June 2022, Paper 3, Q6 (8 marks): In this question you must show detailed reasoning. y y 2 x 9 = + x O – 2 x y 4 e – 1 = 2 x The diagram shows the curves y 2 x + and 9 y 4 e - 1 which intersect
June 2022, AS Paper 1, Q5 (8 marks): d y 2 5 The gradient of a curve is given by d x 3 x . The curve passes through the point (6, 20). = - x (a) Determine the equation of the curve. [4] p 1y (b) He
June 2022, AS Paper 2, Q8 (11 marks): + 8 (a) The quadratic polynomial ax bx , where a and b are constants, is denoted by ( ) f x . Use differentiation from first principles to determine, in terms o
November 2020, AS Paper 1, Q5 (8 marks): The function f is defined by ( ) f x ( x a x )( 3 a x )( b ) where a and b are positive integers. = + + - (a) On the axes in the Printed Answer Booklet, sketch
June 2019, Paper 1, Q11 (10 marks): y x O The diagram shows part of the curve y ln ( x 4 ) . = - y (a) Use integration by parts to show that ln ( x 4 ) d x ( x 4 ) ln x 4 x c . [5] - = - - - + (b)
June 2019, Paper 3, Q6 (10 marks): y x O 2 x 1 - The diagram shows part of the curve y . = 2 x x ( 2 3 )( 1 ) + + Find the exact area of the shaded region, giving your answer in the form p q ln r
June 2019, AS Paper 2, Q7 (8 marks): y y ( 5 x x )( 1 ) = - - x a O y x x x a The diagram shows part of the curve ( 5 )( 1 ) and the line . = - - = Given that the total area of the regions shaded i
June 2018, AS Paper 2, Q8 (9 marks): In this question you must show detailed reasoning. 7 1 The diagram shows part of the graph of y 2 x . The shaded region is enclosed by the curve, the x-axis = -

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