hey asked in Science & MathematicsMathematics · 4 months ago

Calculus I homework question?

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. and sketch the region

y = 7sqrt(64 − x^2), y = 0, x = 1, x = 3;

V = ___ about the x-axis

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  • Ian H
    Lv 7
    4 months ago
    Best Answer

    This might help you to see how to set up the integral.

    An element of volume is created by rotating a rectangle y*dx around x axis.

    That volume element will be coin shaped looked at edge on, although usually called a disc, and should be seen as a stubby cylinder with volume like πr^2*h.

    πr^2 = πy^2, and the height/depth h is taken as dx for an element of volume.

    Since y^2 = 49(64 – x^2), the integrand becomes 49π(64 – x^2)dx

    The x limits go from 1 to 3, so you get

    V = {x: 1 to 3} 49π∫(64 – x^2)dx

    You should find ∫(64 – x^2)dx quite easy

    it is 64x – x^3/3, so applying those limits

    V = 49π[192 – 9 – (64 – 1/3)] = 49π *359/3 = 17542π/3 ~ 18369.9

  • Mike G
    Lv 7
    4 months ago

    V =

    3

    ∫ πy^2 dx = ∫π*49(64-x^2 dx [1,3]

    1

    = π[3136x-49x^3/3] [1,3]

    = π([3136*3-9*49] - [3136-49/3])

    = π(8967-3136+49/3)

    17542π/3

  • 4 months ago

    Have you not been taught the method of disks and washers?

    The area of some disk is pi [ (7sqrt(64 − x^2))^2 - 0^2 ] so that's what we will integrate

    pi * int[1 to 3] 49(64-x^2) dx

    = ?

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