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

Volumes and areas: harder questions Problem set A Answers

Skills

    1. (a) a = 6
    2. (b) b = \(\sqrt{6}\) = 2.45
    1. (a) x = −4
    2. (b) \(50 \pi\) = 157
    1. (a) \(36\)
    2. (b) \(\frac{1296 \pi}{5}\) = 814
    1. (a) \(6 \pi\) = 18.8
    2. (b) V = 950
    1. (a) A curve from the origin rising to (8, 4.00), getting flatter
    2. (b) \(64 \pi\) = 201
    3. (c) \(\frac{1024 \pi}{5}\) = 643
    1. (a) \(\frac{\pi^2}{6}\)
    2. (b) \(\frac{\pi^2}{8}-\frac{\pi}{4}\)

Two curves and technology

    1. (a) (0, 0) and (2, 4)
    2. (b) \(\frac{48 \pi}{5}\) = 30.2 cm³
    1. (a) (0, 0) and (3, 18)
    2. (b) \(27 \pi\) = 84.8
    1. (a) A(0, 3), B(3, 0)
    2. (b) 7.93
    3. (c) 72.8
    1. (a) 10 cm
    2. (b) 503 cm³
    3. (c) 126 cm³

Stretch: unknown shapes and proofs

    1. (a) \(x^2=\sqrt[3]{(y^2-27)^2}\)
    2. (b) 13.1
    3. (c) \(\frac{243 \pi}{4}\) = 191
    4. (d) 109
    1. (a) \(3(a-1)\)
    2. (b) \(\int_1^a 3\ln x\,dx=3a\ln a-3a+3\)
    1. (a) \(y=\frac{r}{h}x\)
    2. (b) \(V=\frac13\pi r^2h\)
    3. (c) \(245 \pi\) cm³ = 770 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) \(\frac{416 \pi}{3}\) cm³ = 436 cm³
    1. (a) A : B = 1 : 4
    1. (a) \(y=\ln(x+1)\)
    2. (b) \(\frac{\pi}{2}\left(e^4-4e^2+7\right)\) = 50.3
    1. (a) A wave between y = 1 and y = 3, with one full period and no part below the x-axis
    2. (b) \(V=\frac{\pi}{2}\int_0^{2\pi}\left(9+8\sin x-\cos 2x\right)dx\)
    3. (c) \(9 \pi^{2}\)

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