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प्रश्न
The region enclosed between the graphs of y = x and y = x2 is denoted by R. Find the volume generated when R is rotated through 360° about x-axis
उत्तर
The region to be revolved is sketched.
Find the intersecting point of y = x and y = x2
x2 = x
x2 – x = 0
x(x – 1) = 0
x = 0, x = 1
If x = 0, y = 0, x = 1, y = 1
∴ Points of intersection are (0, 0), (1, 1)
Volume V = `pi int_0^1 [x^2 - (x^2)^2] "d"x`
= `pi int_0^1 [x^2 -x^4] "d"x`
= `pi[x^3/3 - x^5/5]_0^1`
= `pi[1/3 - 1/5]`
= `pi[(5 -3)/15]`
=`(2pi)/15`
Required volume = `(2pi)/15` cubic units
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