What is the net area between #f(x) = sqrt(x+3)x^3+6x # and the xaxis over #x in [2, 4 ]#?
Thus, the area here is equivalent to:
The integral can be split up:
Bring out multiplicative constants:
Evaluate:
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To find the net area between ( f(x) = \sqrt{x+3}  x^3 + 6x ) and the xaxis over ( x ) in the interval ([2, 4]), follow these steps:
 Identify the intervals where ( f(x) ) is above the xaxis (positive) and below the xaxis (negative).
 Integrate the absolute value of ( f(x) ) over these intervals.
 Subtract the integral of ( f(x) ) where it is below the xaxis from the integral where it is above the xaxis.
The net area can be calculated using definite integrals.
Let's proceed with the calculations:

Identify where ( f(x) ) is positive and negative in the interval ([2, 4]):
Evaluate ( f(x) ) at the interval endpoints:
 ( f(2) = \sqrt{2 + 3}  2^3 + 6(2) = \sqrt{5}  8 + 12 = \sqrt{5} + 4 )
 ( f(4) = \sqrt{4 + 3}  4^3 + 6(4) = \sqrt{7}  64 + 24 = \sqrt{7}  40 )
Since ( f(2) > 0 ) and ( f(4) < 0 ), ( f(x) ) is positive on the interval ( [2, c] ) and negative on the interval ( [c, 4] ), where ( c ) is the root of ( f(x) ) in the interval ([2, 4]).

Find the root ( c ) of ( f(x) ) in the interval ([2, 4]):
Set ( f(x) = 0 ) and solve for ( x ): [ \sqrt{x + 3}  x^3 + 6x = 0 ]
There may not be a simple analytical solution to this equation. You can use numerical methods like the bisection method or Newton's method to approximate the root ( c ).

Integrate the absolute value of ( f(x) ) over the intervals ( [2, c] ) and ( [c, 4] ):
[ \text{Net area} = \int_{2}^{c} f(x) , dx  \int_{c}^{4} f(x) , dx ]
Evaluate these integrals numerically once you have found the value of ( c ).
This procedure will give you the net area between ( f(x) ) and the xaxis over the interval ([2, 4]).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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