How to find the volume of the region enclosed by the curves y = 2 x , y = x 2 rotated about the x-axis?

Question

How to find the volume of the region enclosed by the curves       y    =    2    x  ,       y    =          x      2       rotated about the x-axis?

Quincy Doyle · Accepted Answer

When we rotate the area between these two curves about the x-axis, we get something that looks like a cone... a hollow cone with a curved inside. Consider taking a cross section parallel to the y-z plane and cutting the cone down the middle. The cross-section is going to look like a washer, with an inner radius equal to the height of       y    =          x      2       wherever we are in the solid, and the outer radius equal to       y    =    2    x   at that same x.For reference, the area of a washer is equal to       π    ⋅                  (                  r                      outer                          )            2        -    π    ⋅                  (                  r                      inner                          )            2      , if             r              outer             is the outer radius and             r              inner             is the inner radius. This fact should be immediately obvious - we&#039;re simply subtracting the area of one circle from the area of another.To calculate the volume of the solid, add (integrate) the areas of each cross-sectional washer in the solid. This method is also known as the washers method.Thus, we&#039;ll use the method of washers to find the volume of this solid.The general formula is      V    =                  ∫        a        b              π                  (                  f                      (            x            )                          )            2              d      x        -                  ∫        a        b              π                  (                  g                      (            x            )                          )            2              d      x      where f(x) is a function giving the outer radius of the washer at any x, and g(x) is a function giving the inner radius of the washer. Note that the integrands here represent the areas of circles - and together, they give the area of each infinitesimal washer.In our case, f(x) is equal to 2x and g(x) is equal to             x      2      .      V    =                  ∫        a        b              π                  (        2        x        )            2              d      x        -                  ∫        a        b              π                  (                  x          2                )            2              d      x      Next, we need to worry about the limits of the integrals. Since we&#039;re only concerned with rotating the region between             x      2       and 2x, the limits should involve the locations where       y    =          x      2       and       y    =    2    x   intersect. Intersection occurs at       x    =    0   and       x    =    2  .      V    =                  ∫        0        2              π                  (        2        x        )            2              d      x        -                  ∫        0        2              π                  (                  x          2                )            2              d      x      The difficult part - setting up the appropriate integral with correct limits - is done, so I won&#039;t walk through evaluating the integral. It evaluates to                     64        π            15      , but if you don&#039;t trust me you can evaluate it yourself as an exercise.

How do you find the volume of the region enclosed by the curves y=2x, y=x^2 rotated about the x-axis?

Answered question

Answer & Explanation

New Questions in Calculus 2