Evaluate the Integral

Problem

(∫_0^2)(3*x2*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is 3*x2 and the interval is [0,2]

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

3. Find the antiderivative of 3*x2

(∫_^)(3*x2*d(x))=3⋅(x3)/3

(∫_^)(3*x2*d(x))=x3

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

[x3]20=2−0

5. Simplify the numerical expression to find the final value.

8−0=8

Final Answer

(∫_0^2)(3*x2*d(x))=8

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