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How do you find the volume bounded by x = 1, x = 2, y = 0, and #y = x^2 # revolved about the x-axis?

Answer 1

Evelyn Agard

Volume bounded by #x=1#, #x=2#, #y=0# and #y=x^2# is #(31pi)/5# units.

The area bounded by #x=1#, #x=2#, #y=0# and #y=x^2# is shown below (shaded in grey(

As it revolves around #x#-axis, its volume is given by

#int_1^2(piy^2)dx#

= #int_1^2(pi(x^2)^2)dx#

= #piint_1^2x^4dx#

= #pi[x^5/5]_1^2#

= #pi(2^5/5-1^5/5)#

= #pixx31/5#

= #(31pi)/5#

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Answer 2

Cameron Alexander

To find the volume bounded by ( x = 1 ), ( x = 2 ), ( y = 0 ), and ( y = x^2 ) revolved about the x-axis, you would use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:

[ V = 2\pi \int_{a}^{b} x \cdot f(x) , dx ]

Where ( f(x) ) represents the height of the shell at the given ( x )-value.

In this case, ( a = 1 ), ( b = 2 ), and ( f(x) = x^2 ).

So, the integral to find the volume becomes:

[ V = 2\pi \int_{1}^{2} x \cdot (x^2) , dx ]

Solve this integral to find the volume.

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Answer from HIX Tutor

When evaluating a one-sided 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.