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Use property 8 to estimate the value of the integral ∫ 3

12
x
8
dx. ≤∫ 3
12
x
8
dx≤

User Tim Kane
by
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1 Answer

3 votes

To solve this integral, we can use the power rule for antiderivatives. This states that:

∫x^n dx = x^(n+1) / (n+1) + C, where C is the constant of integration.

The power rule allows us to find the antiderivative (the indefinite integral) of x^8, which is x^9 / 9.

Now, we have to account for the limits of the integral, which are 3 and 12. This is done by substituting these values into the antiderivative:

First plug in the upper limit:
12^9 / 9 = 564697676064 / 9.

Then plug in the lower limit:
3^9 / 9 = 19683 / 9.

To get the final result, the value from the lower limit is subtracted from the value from the upper limit, due to the properties of definite integrals:

(564697676064 / 9) - (19683 / 9) = 573306741.

Therefore, the value of the integral from 3 to 12 of x^8 dx is 573306741.

User Johrn
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