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

AFind 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 ...?

Verified Answer :

syed514

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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