How do I find the volume of the solid generated by revolving the region bounded by #y=x^2#, #y=0#, and #x=2# about the #x#axis? The #y#axis?
1)
2)
the rose region is revolving about the xaxis and yaxis
1)when the shaded region revolving a bout xaxis
2)when the shaded region revolving about the yaxis
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To find the volume of the solid generated by revolving the region bounded by ( y = x^2 ), ( y = 0 ), and ( x = 2 ) about the xaxis and the yaxis, you can use the method of cylindrical shells for the xaxis and the method of disks or washers for the yaxis.

Around the xaxis: Using cylindrical shells, the volume ( V ) is given by: [ V = \int_{0}^{2} 2\pi x \cdot x^2 , dx ] Evaluate this integral to find the volume.

Around the yaxis: Using disks or washers, the volume ( V ) is given by: [ V = \pi \int_{0}^{4} x^2 , dy ] Here, ( x^2 ) represents the radius of the disk or washer, and ( dy ) represents an infinitesimally small change in the height. First, express ( x ) in terms of ( y ) and then evaluate the integral to find the volume.
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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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