How do you evaluate the definite integral #int(5x^(1/3))dx# from #[2,2]#?
where the limits applied to integral come from the interval you have been asked to evaluate. To begin simply integrate the function:
Now evaluate by simply substituting in the limits like so:
It is also possible to arrive at this result intuitively by exploiting the symmetry of the function.
graph{5x^(1/3) [18.19, 18.19, 9.1, 9.09]}
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To evaluate the definite integral ( \int_{2}^{2} 5x^{1/3} , dx ):
 Integrate ( 5x^{1/3} ) with respect to ( x ) to find the antiderivative.
 Evaluate the antiderivative at the upper and lower limits of integration.
 Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the definite integral.
Here are the steps in detail:

Integrate ( 5x^{1/3} ) with respect to ( x ): [ \int 5x^{1/3} , dx = \frac{5}{4}x^{4/3} + C ]

Evaluate the antiderivative at the upper and lower limits of integration: [ \left[\frac{5}{4}x^{4/3}\right]_{2}^{2} ]

Evaluate the antiderivative at ( x = 2 ): [ \frac{5}{4}(2)^{4/3} = \frac{5}{4}(2^{4/3}) ]

Evaluate the antiderivative at ( x = 2 ): [ \frac{5}{4}(2)^{4/3} = \frac{5}{4}((2)^{4/3}) ]

Subtract the value at the lower limit from the value at the upper limit: [ \left[\frac{5}{4}(2^{4/3})\right]  \left[\frac{5}{4}((2)^{4/3})\right] ]
[ = \frac{5}{4}(2^{4/3})  \frac{5}{4}((2)^{4/3}) ]
[ = \frac{5}{4}(2^{4/3}  (2)^{4/3}) ]
[ = \frac{5}{4}(2^{4/3}  2^{4/3}) ]
[ = \frac{5}{4}(0) ]
[ = 0 ]
So, the value of the definite integral ( \int_{2}^{2} 5x^{1/3} , dx ) is ( 0 ).
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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