Using the definition of the integral, find ∫ 0 a x 2 d x

Question

Using the definition of the integral, find       ∫    0    a        x    2    d  x

Erika Gomez · Accepted Answer

As you have that       x    2   is continuous all partitions will give the same integral. Hence      ∫    0    a        x    2    =      lim          n      →      ∞            ∑          i      =      0        n        a    n    ⋅            (      a      i              )        2                    n      2        =      lim          n      →      ∞            a          n      3            ∑          i      =      0        n    (  a  i      )    2    =      lim          n      →      ∞                  a      3              n      3        ⋅            n      ⋅      (      n      +      1      )      ⋅      (      2      n      +      1      )        6  The limit of this is            a      3        3  which we expected.We used that      ∑          i      =      1        n        i    2    =            n      ⋅      (      n      +      1      )      ⋅      (      2      n      +      1      )        6  To see that this partiation is allowed we can use that       x    2   is strict monoton increasing, So      lim          n      →      ∞            ∑          i      =      1        n        a    n              (              i        n            )        2    ≤      ∫    0    a        x    2          d    x  ≤      lim          n      →      ∞            ∑          i      =      0        n              (              i        n            )        2

Using the definition of the integral, find int_0^a x^2dx

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