How do you find the area bounded by #y=x#, #y=1/x^2# the x axis, and x=3?
Start by solving the inequality:
so we can see that in the interval
The required area can be calculated as the definite integral of the bounding curve and therefore is:
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To find the area bounded by the curves ( y = x ), ( y = \frac{1}{x^2} ), the xaxis, and ( x = 3 ), you need to integrate the absolute difference between the two curves from their intersection points to ( x = 3 ).
First, find the intersection point by setting the two equations equal to each other: ( x = \frac{1}{x^2} ). Solve for ( x ) to get ( x^3 = 1 ), which gives ( x = 1 ).
Then, integrate the absolute difference between the two functions from ( x = 1 ) to ( x = 3 ):
[ \text{Area} = \int_{1}^{3} \left (x)  \left( \frac{1}{x^2} \right) \right , dx ]
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To find the area bounded by ( y = x ), ( y = \frac{1}{x^2} ), the xaxis, and ( x = 3 ):

Determine the points of intersection of the curves ( y = x ) and ( y = \frac{1}{x^2} ) by setting them equal to each other and solving for ( x ).

Identify the region of interest based on these points of intersection and the given boundaries.

Integrate the difference of the upper curve and the lower curve within the specified bounds to find the area.
[ \text{Area} = \int_{a}^{b} (f(x)  g(x)) , dx ]
Where ( f(x) ) is the upper curve and ( g(x) ) is the lower curve within the given bounds ([a, b]).
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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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