How do you evaluate the definite integral #int x^(3) dx # from #[b, 2b]#?
I found:
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To evaluate the definite integral ( \int_{b}^{2b} x^3 , dx ), you need to integrate the function ( x^3 ) with respect to ( x ) and then evaluate the result at the upper and lower limits of integration, which are ( 2b ) and ( b ) respectively.
The integral of ( x^3 ) is ( \frac{x^4}{4} ).
So, integrating ( x^3 ) from ( b ) to ( 2b ) gives:
[ \int_{b}^{2b} x^3 , dx = \left[ \frac{x^4}{4} \right]_{b}^{2b} ]
[ = \frac{(2b)^4}{4}  \frac{b^4}{4} ]
[ = \frac{16b^4}{4}  \frac{b^4}{4} ]
[ = 4b^4  \frac{b^4}{4} ]
[ = \frac{15b^4}{4} ]
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To evaluate the definite integral ( \int_{b}^{2b} x^3 , dx ), you can follow these steps:

Integrate the function ( x^3 ) with respect to ( x ): ( \int x^3 , dx = \frac{x^4}{4} + C ), where ( C ) is the constant of integration.

Evaluate the definite integral by substituting the upper and lower limits of integration: [ \left[ \frac{x^4}{4} \right]_{b}^{2b} ]

Substitute the upper limit (2b) into the integrated function: [ \frac{(2b)^4}{4} = \frac{16b^4}{4} = 4b^4 ]

Substitute the lower limit (b) into the integrated function: [ \frac{b^4}{4} ]

Evaluate the definite integral by subtracting the result at the lower limit from the result at the upper limit: [ 4b^4  \frac{b^4}{4} ] [ = \frac{16b^4}{4}  \frac{b^4}{4} ] [ = \frac{15b^4}{4} ]
So, the value of ( \int_{b}^{2b} x^3 , dx ) from ( b ) to ( 2b ) is ( \frac{15b^4}{4} ).
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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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