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

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

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The diagram shows the graph of \(y = x^{2} + 2 x - 15\). The shaded region is bounded by the curve, the x-axis and the lines \(x = -2\) and \(x = 2\). Find the area of the shaded region.

    −2−112−15−10−5xy

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

    −2−112345−15−10−5xy

  3. The diagram shows the graph of \(y = x^{4} - 7 x^{2} + 6 x\) between \(x = -3\) and \(x = 2\). Use your calculator to find the total area of the shaded regions.

    −4−3−2−1123−25−20−15−10−55xy

  4. The diagram shows the graph of \(y = \sin x - \cos x\) for \(0 \le x \le 2 \pi\). Use your calculator, in radians, to find the total area between the curve and the x-axis.

    −11234567−2−112xy

  5. Evaluate \(\int_{0}^{4}\sqrt{3x+4}\,dx\).

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

  7. Find the value of a, where \(a > 4\), such that \(\int_{4}^{a} \frac{10}{\sqrt{x}}\,dx = 30\).

  8. A curve has equation \(y = - x^{2} - 2 x + 15\).

    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 drainage channel has a cross-section bounded by the curve \(y = \sqrt{\ln x}\), the x-axis and the lines \(x = 2\) and \(x = 7\), where x and y are in metres. Use your calculator to find the area of the cross-section.

  2. The roof of a tunnel is modelled by the curve \(y = \ln(26 - x^2)\), where x and y are in metres and the x-axis is the road. Use your calculator to find the area of the tunnel's cross-section.

    −6−4−2246−11234xy

  3. The thickness of a snowdrift, \(t\) metres, at a distance \(x\) metres from its crest is modelled by \(t=1.8e^{-x^2/10}\) for \(x\ge 0\). The area of the cross-section from the crest to the point \(x=k\) is 4 m². Find k, to 3 significant figures.

Stretch

  1. The diagram shows the graph of \(y = x^{3} - 4 x^{2} - 7 x + 10\), which crosses the x-axis at \(x = −2, 1, 5\). The shaded region is bounded by the curve and the x-axis.

    −2246−20−1010xy
    1. (a) Evaluate \(\int_{-2}^{1} y\,dx\) and \(\int_{1}^{5} y\,dx\).
    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