How do you use limits to evaluate #int(x^2+4x2)dx# from [1,4]?
Here is a limit definition of the definite integral. (I don't know if it's the one you are using.)
I prefer to do this type of problem one small step at a time.
Evaluate the sums
(We used summation formulas for the sums in this step.)
Rewrite before finding the limit
To finish the calculation, we have
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To evaluate the integral ( \int_{1}^{4} (x^2 + 4x  2) , dx ) using limits, you first find the antiderivative of the function ( x^2 + 4x  2 ), which is ( \frac{1}{3}x^3 + 2x^2  2x ). Then, you evaluate this antiderivative at the upper limit (4) and subtract the result of evaluating it at the lower limit (1). So, ( \int_{1}^{4} (x^2 + 4x  2) , dx = \left[ \frac{1}{3} \cdot 4^3 + 2 \cdot 4^2  2 \cdot 4 \right]  \left[ \frac{1}{3} \cdot 1^3 + 2 \cdot 1^2  2 \cdot 1 \right] ). Calculating this expression gives the value of the definite integral.
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To evaluate the definite integral ( \int_{1}^{4} (x^2 + 4x  2) , dx ) using limits, follow these steps:

First, find the antiderivative of the integrand, which is ( \frac{1}{3}x^3 + 2x^2  2x ).

Evaluate the antiderivative at the upper and lower limits of integration:
 At ( x = 4 ): ( \frac{1}{3}(4)^3 + 2(4)^2  2(4) = \frac{64}{3} + 32  8 = \frac{64}{3} + 24 = \frac{112}{3} )
 At ( x = 1 ): ( \frac{1}{3}(1)^3 + 2(1)^2  2(1) = \frac{1}{3} + 2  2 = \frac{1}{3} )

Subtract the value of the antiderivative at the lower limit from the value at the upper limit: ( \frac{112}{3}  \frac{1}{3} = \frac{111}{3} = 37 )
Therefore, ( \int_{1}^{4} (x^2 + 4x  2) , dx = 37 ).
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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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