Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 9 sin(x), y =

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 9 sin(x), y = 9 cos(x), 0 ≤ x ≤ /4; about y = −1

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Thành Công 3 years 2021-08-31T16:02:50+00:00 1 Answers 13 views 0

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  1. thienthanh
    0
    2021-08-31T16:04:10+00:00
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    Step-by-step explanation:

    A cross-section is a washer with an inner radius of 8sin(x) – (-1) and an outer radius of 8cos(x) – -(1), so its area would be:

    A(x) = π[(8cos(x) + 1)^2 − (8sin(x) + 1)^2]

    = π[64cos^2(x) + 16cos(x) + 1 – 64sin^2(x) − 16sin(x) − 1]

    = π[64cos(2x) + 16cos(x) – 16sin(x)]

    => V(x) = ∫[0,π/4] π[64cos(2x) + 16cos(x) – 16sin(x)] dx

    = π[32sin(2x) + 16sin(x) + 16cos(x)] |[0,π/4]

    = π[32sin(π/2) + 16√2/2 + 16√2/2 – 16]

    = π(32 – 16 + 16√2) = π(16 + 16√2)

    The volume of the region is π(16 + 16√2).

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