The red curve refers to the graph of the first function: `y =xsin(x^2) ` while the the the blue curve refers to the graph of the second function: `y=x^4` .
As shown in the xy-plane, the two graphs intersect , approximately, at the following points: (0,0) and (0.9,0.65).
Based on these intersection points, the limits of integration with respect to x will be from x= 0 to x=0.9.
The formula for the " Area...
The red curve refers to the graph of the first function: `y =xsin(x^2) ` while the the the blue curve refers to the graph of the second function: `y=x^4` .
As shown in the xy-plane, the two graphs intersect , approximately, at the following points: (0,0) and (0.9,0.65).
Based on these intersection points, the limits of integration with respect to x will be from x= 0 to x=0.9.
The formula for the " Area between Two Curves" is:
A= ` int_a^b[f(x)-g(x)]dx`
such that` f(x)gt=g(x) ` on the interval of [a,b].
This is the same as A = `int_a^b[y_(above) - y_(below)]dx`
where the bounded area is in between `y_(above) = f(x) ` and `y_(below)= g(x)` .
Applying the formula on the given problem, the integration will be:
A = `int_0^(0.9)[x*sin(x^2) - x^4]dx`
=` [-(cos(x^2)/2) -x^5/5] |_0^(0.9)`
= `[-(cos((0.9)^2)/2) -(0.9)^5/5]-[-(cos((0)^2)/2) -0^5/5]`
=-0.4628472164 - (-0.5)
= -0.4628472164 + 0.5
= 0.03715278360
`~~` 0.0372 as the Area of the region bounded by the curves shown above.
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