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5 November, 15:40

Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of y=sinx and y=cosx.

a) Find the area of R

b) Find the volume of the solid generated when R is revolved about the x-axis

c) Find the volume of the solid whose base is R and whose cross sections cut by planes perpendicular to the x-axis are squares ... ?

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  1. 5 November, 18:33
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    since sin and cos = each other at pi/4; take your integrals from 0 to pi/4

    [S] cos (t) dt - [S] sin (t) dt; [0, pi/4]

    to revolve it around the x axis;

    we do a sum of areas [S] 2pi [f (x) ]^2 dx

    take the cos first and subtract out the sin next; like cutting a hole out of a donuts.

    pi [S] cos (x) ^2 dx - [S] sin (x) ^2 dx; [0, pi/4]

    cos (2t) = 2cos^2 - 1

    cos^2 = (1+cos (2t)) / 2

    1/sqrt (2) - (-1/sqrt (2) + 1)

    1/sqrt (2) + 1/sqrt (2) - 1

    (2sqrt (2) - sqrt (2)) / sqrt (2) = sqrt (2) / sqrt (2) = 1
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