Evaluate the definite integral ∫(1 to 2) (3x^2)dx.

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Question: Evaluate the definite integral ∫(1 to 2) (3x^2)dx.

Options:

Correct Answer: 6

Solution:

The integral evaluates to 6.

Evaluate the definite integral ∫(1 to 2) (3x^2)dx.

Evaluate the definite integral ∫(1 to 2) (3x^2)dx.

Step 1: Identify the function to integrate, which is 3x^2.
Step 2: Find the antiderivative of 3x^2. The antiderivative is x^3.
Step 3: Evaluate the antiderivative at the upper limit (2). Calculate (2^3) = 8.
Step 4: Evaluate the antiderivative at the lower limit (1). Calculate (1^3) = 1.
Step 5: Subtract the value at the lower limit from the value at the upper limit. So, 8 - 1 = 7.
Step 6: Multiply the result by 3 (the coefficient in front of x^2). So, 3 * 7 = 21.
Step 7: The final answer is 21.

Definite Integral Evaluation – The process of calculating the area under the curve of a function between two specified limits.
Power Rule for Integration – A method used to integrate polynomial functions, where the integral of x^n is (x^(n+1))/(n+1) + C.