How do you find the definite integral of #int (4x^2+2)dx# from #[0,3]#?

Answer 1

Hi!

To solve most definite integral problems, the process is pretty well the same:

1) Integrate the function - that is, get it's "anti-derivative"
2) Evaluate at the upper limit
3) Evaluate at the lower limit
4) Evaluate difference between the upper and lower limit

In your example, you must integrate your function, then evaluate it at 0, then at 3, then evaluate the difference.

I'll run through it below:

#int_0^3(4x^2+2)dx# on # [0, 3]#
#4/3x^3 + 2x -> # Note: you omit the constant of integration "+ c" as it cannot be evaluated without initial conditions.

Also, you'd normally put a vertical bar after the integrated function with the lower limit at the bottom, and the upper limit at the top. This simply means, "Evaluate the function from the lower limit "a" to the upper limit "b". I'd use that notation however, I'm not sure how to put it in, haha! :)

Anyway, now you can evaluate your integrated function at both limits:

# = (4/3(3)^3 + 2(3)) - (4/3(0)^3 + 2(0)) # # = 42 #

That's it! That's how you evaluate definite integrals! Hopefully everything was clear and helpful! :)

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Answer 2

To find the definite integral of the function ( \int (4x^2+2)dx ) from ( x = 0 ) to ( x = 3 ), you first need to integrate the function with respect to ( x ) and then evaluate the result at the upper and lower limits of integration.

The indefinite integral of ( 4x^2+2 ) with respect to ( x ) is ( \frac{4}{3}x^3 + 2x + C ), where ( C ) is the constant of integration.

To find the definite integral from ( x = 0 ) to ( x = 3 ), you evaluate the antiderivative at the upper limit (3) and subtract the antiderivative evaluated at the lower limit (0).

So, ( \int_{0}^{3} (4x^2+2)dx = \left[\frac{4}{3}x^3 + 2x\right]_{0}^{3} ).

Evaluating at the upper limit: [ \left(\frac{4}{3}(3)^3 + 2(3)\right) = \left(\frac{4}{3}(27) + 6\right) = \left(36 + 6\right) = 42 ]

Evaluating at the lower limit: [ \left(\frac{4}{3}(0)^3 + 2(0)\right) = \left(0 + 0\right) = 0 ]

So, ( \int_{0}^{3} (4x^2+2)dx = 42 - 0 = 42 ).

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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