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The integral is $\"\"$ .

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Theorem :

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If $\"\"$ is integrable on $\"\"$ , then $\"\"$ ,

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where width of the interval $\"\"$ and $\"\"$ .

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Here $\"\"$ , $\"\"$ and $\"\"$ .

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Width of the interval is $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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The sum of squares of $\"\"$ natural numbers is $\"\"$ .

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The sum of cubes of $\"\"$ natural numbers is $\"\"$ .

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$\"\"$

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$\"\"$ .