Trig Functions (1 Viewer)

[siraulo23]

1) Find the area bounded by the curve y=sinx and y=cosx in exact form

Thanks in advance!

 

[random-1006]

1) Find the area bounded by the curve y=sinx and y=cosx in exact form

Thanks in advance!

it really should give a domain,

first solve for intersection ( i will take first two positive intersections, and the region between those two will be my area

ok solve for intersections

sinx=cosx
tanx=1 ( related angle pi/4 , in first and third quadrants)

therefore
x= pi/4 and 5pi/4

this will be the area i will do, in the question it should say find the area bounded between certain endpoints, obviously if we have no end points the area is infinite

now draw the two graphs and sinx will be above cosx in that domain ( pi/4 <= x <= 5pi/4)

now the area between two curves is integral ( top y - bottom y ) dx ( between limits)

so integral (sinx-cosx) dx from pi/4 .. 5pi/4
= [-cosx -sinx] pi/4 .. 5pi/4
= - [cosx +sinx] pi/4 .. 5pi/4
= - {[ -1/sqrt2 +-1/sqrt2 ] - [1/sqrt2 +1/sqrt2] }
= - [ -2/sqrt2-2/sqrt2]
= 4/sqrt2 ( rationalise)
= 4 sqrt2 / 2
= 2 (sqrt 2) units^2