Find the area produced by:[math: A=∫16(8x3+12x2+4)dx](square units)

Question

BlackTom AI · Accepted Answer

To start, I’ll rewrite the area integral in a form that’s easy to evaluate: A = ∫ from x=1 to x=6 of (8x^3 + 12x^2 + 4) dx.
Next, I’ll find an antiderivative term by term.
For the 8x^3 term: the antiderivative is 8 · (x^4/4) = 2x^4.
For the 12x^2 term: the antiderivative is 12 · (x^3/3) = 4x^3.
For the 4 term: the antiderivative is 4x.
So the overall antiderivative is F(x) = 2x^4 + 4x^3 + 4x.
Now, evaluate F(x) at the bounds x = 6 and x = 1.
Compute F(6): 2(6^4) + 4(6^3) + 4(6) = 2(1296) + 4(216) + 24 = 2592 + 864 + 24 = 3480.
Compute F(1): 2(1^4) + 4(1^3) + 4(1) = 2 + 4 + 4 = 10.
Subtract the value at the lower bound from the value at the upper bound: A = F(6) − F(1) = 3480 − 10 = 3470.
Therefore, the area produced by the given integral is 3470 square units. If you want to check quickly, you can reassemble the calculation by grouping constants or differentiating the result to see it matches the integrand, but the evaluation above is straightforward and correct.

UCIC 202503 EMTH118 Quiz 11

Find the area produced by:[math: A=∫16(8x3+12x2+4)dx](square units)

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