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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.10

Volumes and areas: harder questions Problem set B Answers

Skills

    1. (a) a = 12
    2. (b) b = \(\sqrt{10}\) = 3.16
    1. (a) x = −6
    2. (b) \(\frac{121 \pi}{2}\) = 190
    1. (a) \(\frac{256}{3}\)
    2. (b) \(\frac{16384 \pi}{15}\) = 3430
    1. (a) \(8 \pi\) = 25.1
    2. (b) V = 724
    1. (a) A curve from the origin rising to (12, 6.00), getting flatter
    2. (b) \(216 \pi\) = 679
    3. (c) \(\frac{3456 \pi}{5}\) = 2170
    1. (a) \(\frac{\pi^2}{8}\)
    2. (b) \(\frac{\pi^2}{12}-\frac{\pi}{6}\)

Two curves and technology

    1. (a) (0, 0) and (3, 9)
    2. (b) \(\frac{729 \pi}{10}\) = 229 mm³
    1. (a) (0, 0) and (4, 48)
    2. (b) \(128 \pi\) = 402
    1. (a) A(0, 4), B(4, 0)
    2. (b) 13.1
    3. (c) 152
    1. (a) 6 cm
    2. (b) 106 cm³
    3. (c) 35.3 cm³

Stretch: unknown shapes and proofs

    1. (a) \(x^2=\sqrt[3]{(y^2-64)^2}\)
    2. (b) 26.9
    3. (c) \(192 \pi\) = 603
    4. (d) 297
    1. (a) \(2(a-1)\)
    2. (b) \(\int_1^a 2\ln x\,dx=2a\ln a-2a+2\)
    1. (a) \(y=\frac{R}{H}x\)
    2. (b) \(V=\frac13\pi R^2H\)
    3. (c) \(108 \pi\) cm³ = 339 cm³
    1. (a) \(x^2+y^2=r^2\)
    2. (b) \(V=\frac{\pi}{3}\left(2r^3-3r^2a+a^3\right)\)
    3. (c) \(63 \pi\) cm³ = 198 cm³
    1. (a) A : B = 1 : 5
    1. (a) \(y=\ln(x+2)\)
    2. (b) \(\frac{\pi}{2}\left(e^4-8e^2+28-8\ln 2\right)\) = 28.2
    1. (a) A wave between y = 2 and y = 4, with one full period and no part below the x-axis
    2. (b) \(V=\frac{\pi}{2}\int_0^{2\pi}\left(19+12\cos x+\cos 2x\right)dx\)
    3. (c) \(19 \pi^{2}\)

jmaths.xyz · AI HL · AHL 3.10 Harder volumes and areas of revolution