How do you find the area of the region bounded by the graph of #f(x)=1x^2# and the #x#axis on the interval #[1,1]# ?
This is a definite integral problem.
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To find the area of the region bounded by the graph of ( f(x) = 1  x^2 ) and the xaxis on the interval ([1, 1]), follow these steps:

Determine the xvalues where ( f(x) = 0 ) to find the bounds of integration.
( 1  x^2 = 0 )
( x^2 = 1 )
( x = \pm 1 )

Set up the definite integral to find the area:
[ \text{Area} = \int_{1}^{1} f(x) , dx ]

Substitute the function ( f(x) = 1  x^2 ) into the integral:
[ \text{Area} = \int_{1}^{1} 1  x^2 , dx ]

Evaluate the integral.
Since the function is symmetric about the yaxis, you can split the integral into two parts:
[ \text{Area} = 2 \int_{0}^{1} (1  x^2) , dx ]

Integrate ( (1  x^2) ) from ( 0 ) to ( 1 ):
[ \text{Area} = 2 \left[ x  \frac{x^3}{3} \right]_{0}^{1} ]
[ \text{Area} = 2 \left[ \left(1  \frac{1}{3}\right)  \left(0  0\right) \right] ]
[ \text{Area} = 2 \left( \frac{2}{3} \right) ]
[ \text{Area} = \frac{4}{3} ]
Therefore, the area of the region bounded by the graph of ( f(x) = 1  x^2 ) and the xaxis on the interval ([1, 1]) is ( \frac{4}{3} ) square units.
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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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