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Question

Find the area of the region bounded by the parabola y2=2px, and x2=2py.

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Solution

We have, y2=2 px and x2=2 py

y=2 px

x2=2p.2px

x4=4p2.(2px)

x4=8p3x

x48p3x=0

x3(x8p3)=0

x=0, 2p

Required area=2p02pxdx2p0x22pdx=2p2p0x12dx12p2p0x2dx=2p[2(x)323]2p012p[x33]2p0=2p[23(2p)320]12p[13(2p)30]=2p(23.22p32)12p(138p3)=2p(423p32)12p(83p3)=423.2p286p2=(168)p26=8p26=4p23sq.units


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