Evaluate the Integral

Problem

(∫_0^2)(x^2*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=x^2 and the interval is [0,2]

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

3. Find the antiderivative of x^2

(∫_^)(x^2*d(x))=(x^3)/3

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

[(x^3)/3]^2_0

5. Substitute the values x=2 and x=0 into the expression.

2^3/3−0^3/3

6. Simplify the arithmetic to find the final value.

8/3−0=8/3

Final Answer

(∫_0^2)(x^2*d(x))=8/3

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