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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.5

Name Date

Area between a curve and the x-axis Problem set A

Sketch or read the diagram first. Work by hand where the question asks for exact working, and use the calculator for the others. Give decimal answers to 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. The diagram shows the graph of \(y = x^{2} - 4 x - 5\). The shaded region is bounded by the curve, the x-axis and the lines \(x = 0\) and \(x = 4\). Find the area of the shaded region.

    1234−10−8−6−4−2xy

  2. The diagram shows the graph of \(y = x^{3} - 4 x^{2} + 3 x\). The curve crosses the x-axis at \(x = 0, 1, 3\). Find the total area of the shaded region between \(x = 0\) and \(x = 3\).

    −0.50.511.522.533.5−3−2−11xy

  3. The diagram shows the graph of \(y = x^{4} - 2 x^{3} - 5 x^{2} + 6 x\) between \(x = -2\) and \(x = 3\). Use your calculator to find the total area of the shaded regions.

    −3−2−11234−10−8−6−4−22xy

  4. The diagram shows the graph of \(y = \sin x + \cos x\) for \(\frac{\pi}{2} \le x \le 2 \pi\). Use your calculator, in radians, to find the total area between the curve and the x-axis.

    1234567−2−11xy

  5. Evaluate \(\int_{1}^{5}\sqrt{4x+5}\,dx\).

  6. The curve \(y = 4\sin 5x\) crosses the x-axis at \(x = 0\) and \(x = \frac{\pi}{5}\). Find the exact area enclosed by this part of the curve and the x-axis.

  7. Find the value of a, where \(a > 9\), such that \(\int_{9}^{a} \frac{6}{\sqrt{x}}\,dx = 18\).

  8. A curve has equation \(y = - x^{2} + 2 x + 8\).

    1. (a) Find the x-coordinates of the points where the curve crosses the x-axis.
    2. (b) Find the area enclosed by the curve and the x-axis.

In context

  1. A flower bed has a curved edge modelled by \(y = \sqrt{\ln x}\), where x and y are in metres. The bed is bounded by this curve, the x-axis and the lines \(x = 3\) and \(x = 9\). Use your calculator to find the area of the flower bed.

  2. The arch of a footbridge is modelled by the curve \(y = \ln(17 - x^2)\), where x and y are in metres and the x-axis is the ground. Use your calculator to find the area of the opening under the arch.

    −4−224−1123xy

  3. The depth of a canal, \(d\) metres, at a distance \(x\) metres from its centre line is modelled by \(d=1.5e^{-x^2/12}\) for \(x\ge 0\). The cross-sectional area of water from the centre line to the point \(x=k\) is 3.5 m². Find k, to 3 significant figures.

Stretch

  1. The diagram shows the graph of \(y = x^{3} + 2 x^{2} - 5 x - 6\), which crosses the x-axis at \(x = −3, −1, 2\). The shaded region is bounded by the curve and the x-axis.

    −4−3−2−1123−10−55xy
    1. (a) Evaluate \(\int_{-3}^{-1} y\,dx\) and \(\int_{-1}^{2} y\,dx\).
    2. (b) Hence find the total area of the shaded region.

jmaths.xyz · AI HL · AHL 3.5 Area between a curve and the x-axis