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 x-axis, 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 x-axis, and ( x = 3 ):
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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 ).
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Identify the region of interest based on these points of intersection and the given boundaries.
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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 one-sided 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 one-sided 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 one-sided 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 one-sided 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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