Evaluate the Integral

Problem

(∫_−1^2)(9−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 9−x2

F(x)=9*x−(x3)/3

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and the lower limit −1

(∫_−1^2)(9−x2*d(x))=[9*x−(x3)/3]2(−1)

5. Substitute the upper limit into the antiderivative.

9*(2)−2/3=18−8/3

6. Substitute the lower limit into the antiderivative.

9*(−1)−((−1)3)/3=−9+1/3

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

(18−8/3)−(−9+1/3)

8. Simplify the expression by combining the terms.

18−8/3+9−1/3

27−9/3

27−3=24

Final Answer

(∫_−1^2)(9−x2*d(x))=24

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