Evaluate the Integral

Problem

(∫_−2^3)(36−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 36−x2

(∫_^)(36−x2*d(x))=36*x−(x3)/3

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 3 and the lower limit −2

[36*x−(x3)/3]3(−2)

5. Substitute the upper limit into the antiderivative.

36*(3)−3/3=108−9=99

6. Substitute the lower limit into the antiderivative.

36*(−2)−((−2)3)/3=−72−(−8/3)=−72+8/3

7. Simplify the lower limit value by finding a common denominator.

−72+8/3=−216/3+8/3=−208/3

8. Subtract the lower limit value from the upper limit value.

99−(−208/3)=99+208/3

9. Calculate the final result.

297/3+208/3=505/3

Final Answer

(∫_−2^3)(36−x2*d(x))=505/3

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