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IB Mathematics: Applications and Interpretation HL · 13.5: Integration · 13.5.4

Name Date

Definite integrals and area under a curve Problem set A

Write the definite integral first. Then evaluate it on the calculator and give 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. Evaluate \(\int_{-2}^{1} 5x^4\,dx\).

  2. Evaluate \(\int_{1}^{3} (4-5t^3)\,dt\).

  3. Evaluate \(\int_{2}^{5} 8x^{-3}\,dx\).

  4. Evaluate \(\int_{1}^{3} (2x-1)^3\,dx\).

  5. The diagram shows the curve y = 12 − x². Find the area of the shaded region between x = 1 and x = 3.

    −1123424681012xyy = 12 − x²

  6. The diagram shows the curve y = 3 + 6/x². Find the area of the shaded region between x = 1 and x = 3.

    0.511.522.533.5424681012xyy = 3 + 6/x²

  7. The diagram shows the curve y = 6/x. Find the area of the shaded region between x = 1 and x = 5.

    123456246810xyy = 6/x

  8. The diagram shows the curve \(y=\frac{x^2}{4}\) inside the rectangle with corners \((0,0)\), \((6,0)\), \((6,9)\) and \((0,9)\). Find the area of the shaded region above the curve.

    2468246810xyy = x²/4y = 9

In context

  1. A flower bed lies between the x-axis and the curve \(y=5-0.25x^2\), from \(x=0\) to \(x=4\). Lengths are in metres. Find the area of the flower bed.

    −112345−1123456xyy = 5 − 0.25x²

  2. A medicine is absorbed into the blood at a rate of \(40e^{-0.25t}\) milligrams per hour, where t is the time in hours after the dose. Find the total amount absorbed in the first 6 hours.

  3. The cross-section of a tunnel is the region between the x-axis (the ground) and the curve \(y=8-0.5x^2\), where x and y are in metres. The tunnel is 150 m long.

    −6−4−22462468xyy = 8 − 0.5x²
    1. (a) Find the width of the tunnel at ground level.
    2. (b) Find the area of the cross-section.
    3. (c) Find the volume of rock removed to make the tunnel.

Stretch

  1. The diagram shows the curve \(y=5-\dfrac{2}{x^2}\). It meets the x-axis at the point \((a,0)\).

    0.511.522.533.54−22468xyy = 5 − 2/x²
    1. (a) Find the exact value of a.
    2. (b) Find the area under the curve from x = a to x = 3.

jmaths.xyz · AI HL · 13.5.4 Definite integrals and area under a curve