Evaluate the Integral

Problem

(∫_0^4)(4*x−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 each term in the integrand.

3. Find the antiderivative of the expression 4*x−x2

(∫_^)(4*x−x2*d(x))=(4*x2)/2−(x3)/3

4. Simplify the antiderivative expression.

2*x2−(x3)/3

5. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 4 and the lower limit 0

(2*(4)2−4/3)−(2*(0)2−0/3)

6. Calculate the values within the parentheses.

(2*(16)−64/3)−(0−0)

7. Subtract the terms to find the final numerical value.

32−64/3

8. Find a common denominator to perform the subtraction.

96/3−64/3=32/3

Final Answer

(∫_0^4)(4*x−x2*d(x))=32/3

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