# Y=sqrt(x), y=0, x=0,and x=2 a. Find the area of the region b. find the volume of the solid formed by rotating the region about the x-axis c. find the volume of the solid?

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a. The area of the region is given by the integral of ( y = \sqrt{x} ) from ( x = 0 ) to ( x = 2 ): [ \text{Area} = \int_{0}^{2} \sqrt{x} , dx ]

b. To find the volume of the solid formed by rotating the region about the x-axis, we use the method of cylindrical shells. The volume is given by the integral: [ V = 2\pi \int_{0}^{2} x \sqrt{x} , dx ]

c. To find the volume of the solid formed by rotating the region about the y-axis, we need to use the washer method. The volume is given by the integral: [ V = \pi \int_{0}^{2} (\sqrt{x})^2 , dx ]

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