A-level · Integration

Definite integration and area under a curveExam-style questions

Finding the area of a region with a curved edge. Thin strips and the limit of a sum; why an antiderivative measures area (the fundamental theorem); evaluating with square brackets, top limit minus bottom limit, with no + C; why a region below the x-axis counts as negative, so an integral and an area can differ; splitting and reversing the limits; and finding a limit that is a letter.

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5 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    The curve y=x(4−x)y = x(4 - x) and the line y=xy = x enclose a finite region.
    (a) Find the xx-coordinates of the points where they meet.
    (b) Find the area of the region enclosed between them.
    Level 3 · 5 marksTry it
  2. 2
    The region RR is bounded by the curve y=41+x4y = \dfrac{4}{1 + x^4}, the xx-axis, the yy-axis and the line x=1x = 1. Priya estimates the area of RR by splitting the interval 0≤x≤10 \le x \le 1 into strips of width δx=0.2\delta x = 0.2. On each strip she draws two rectangles, one lying entirely below the curve and one whose top lies entirely above the curve, as shown in the diagram.0.20.40.60.81xy(a) Show that Priya's rectangles give a lower bound of 3.253.25 and an upper bound of 3.653.65 for the area of RR, correct to 3 significant figures.

    (1 mark)
    (b) Priya repeats the process, halving the width of the strips each time. Her lower and upper bounds, correct to 3 significant figures, are given in the table.
    Width of strip, δx\delta x0.20.20.10.10.050.050.0250.025
    Lower bound3.253.253.363.363.423.423.443.44
    Upper bound3.653.653.563.563.523.523.493.49
    Use the table to estimate the area of RR correct to 2 significant figures, giving a reason for your estimate.

    (2 marks)
    (c) Write down an expression in yy and δx\delta x that gives the area of RR exactly.

    (2 marks)
    Level 3 · 5 marksTry it