Evaluate the integral?

Question

#intRint siny/y dA#, R;the region bounded by #y=x and y=sqrtx#

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Answer 1 · 2018-06-27T08:44:04

#= 1 - sin 1 #

#intint_R siny/y dA#

#= int_0^1 dx int_x^sqrtx dy qquad siny/y#

#= int_0^1 dy int_(y^2)^y dx qquad siny/y#

The second one looks more promising:

#= int_0^1 dy qquad [ xsiny/y]_(x=y^2)^y#

#= int_0^1 dy qquad siny -y siny #

# = [(y - 1) cosy - siny]_0^1 = 1 - sin 1 #

graph{(y - sqrtx)(y-x)=0 [-1.502, 1.5, -0.25, 1.517]}