Evaluate the Integral

Problem

(∫_−4^4)(16−x2*d(x))

Solution

1. Identify the integral as a definite integral of a polynomial function.

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) to find the antiderivative.

3. Find the antiderivative of the expression 16−x2

(∫_^)(16−x2*d(x))=16*x−(x3)/3

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 4 and the lower limit −4

[16*x−(x3)/3]4(−4)

5. Substitute the upper limit into the antiderivative.

16*(4)−4/3=64−64/3

6. Substitute the lower limit into the antiderivative.

16*(−4)−((−4)3)/3=−64−(−64)/3=−64+64/3

7. Subtract the lower limit evaluation from the upper limit evaluation.

(64−64/3)−(−64+64/3)

8. Simplify the expression by combining like terms.

64−64/3+64−64/3

128−128/3

9. Calculate the final numerical value using a common denominator.

384/3−128/3=256/3

Final Answer

(∫_−4^4)(16−x2*d(x))=256/3

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