Need help with integration (1 Viewer)

[Tabris]

volume bounded by X^3 and X^2 and the x - axis rotated on the x -axis

 

[CM_Tutor]

The curves y = x² and y = x³ meet at two points, (0, 0) and (1, 1). To find the volume formed when an area between two curves is rotated about an axis, you must find the individual areas separately. That is,

V₁ = int (from 0 to 1) pi * y² dx, where y = x²
= pi * int (from 0 to 1) x⁴ dx
= pi * [x⁵ / 5] (from 0 to 1)
= (pi / 5) * [(1)⁵ - (0)⁵]
= pi / 5 cu units

V₂ = int (from 0 to 1) pi * y² dx, where y = x³
= pi * int (from 0 to 1) x⁶ dx
= pi * [x⁷ / 7] (from 0 to 1)
= (pi / 7) * [(1)⁷ - (0)⁷]
= pi / 7 cu units

Now, V_TOT = V₁ - V₂ = (pi / 5) - (pi / 7) = pi * (7 - 5) / (7 * 5)
So, V_TOT = 2 * pi / 35 cu units

 

[Tabris]

thx teach, i know where i went wrong

 

[Tabris]

a tricky 2 unit question

Volume bounded by Y = x^2 and Y = (x+2)^2 and the x -axis, rotating on the x axis.

This is where i am up to

x^2 = x^2+4x+4

only 1 common point (-1,1)

i am stuck here, anything i missed or any mistakes?

 

[CM_Tutor]

Draw a diagram. The area that you are trying to rotate about the x-axis is bounded by the x-axis (from x = -2, the x-intecept of y = (x + 2)² to x = 0, the x-intercept of y = x²), the curve y = (x + 2)² from x = -2 to the point of intersection at x = -1, and the curve y = x² from the point of intersection at x = -1 to x = 0. You have to rotate the two areas around the axis separately. Thus, V_TOT = V₁ + V₂ where:

V₁ = int (from -2 to -1) pi * y² dx where y = (x + 2)²

and

V₂ = int (from -1 to 0) pi * y² dx where y = x²

Note that symmetry will ensure that V₁ = V₂ = pi / 5 cu units, and so V_TOT = 2 * pi / 5 cu units