1.

Find the area of the region in the first quadrant enclosed by the y-axis,the line `y=x`and the circle `x^2+y^2=32 ,`using integration.

Answer» `r=sqrt32=4sqrt2`
`y=r=4sqrt2`
`2y^2=32`
`y^2=16`
`y=4`
`area=int_0^4sqrt(32-x^2)-xdx`
`=int_0^4sqrt((4R)^2-x^2)dx-int_0^4xdx`
`=(4pi-8)-[x^2/2]_0^4`
`=4pi-8-[(16-0)/2]`
`=4pi-8-8`
`=4pi-16`.


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