Evaluate ∫_1^2 (3x^2 - 4) dx.

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Question: Evaluate ∫_1^2 (3x^2 - 4) dx.

Options:

Correct Answer: 1

Solution:

The integral evaluates to [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.

Evaluate ∫_1^2 (3x^2 - 4) dx.

Evaluate ∫_1^2 (3x^2 - 4) dx.

Step 1: Identify the integral to evaluate: ∫_1^2 (3x^2 - 4) dx.
Step 2: Find the antiderivative of the function 3x^2 - 4. The antiderivative is x^3 - 4x.
Step 3: Write the antiderivative with limits: [x^3 - 4x] from 1 to 2.
Step 4: Calculate the value of the antiderivative at the upper limit (x = 2): (2^3 - 4*2) = (8 - 8) = 0.
Step 5: Calculate the value of the antiderivative at the lower limit (x = 1): (1^3 - 4*1) = (1 - 4) = -3.
Step 6: Subtract the lower limit result from the upper limit result: 0 - (-3) = 0 + 3 = 3.
Step 7: Conclude that the value of the integral is 3.

Definite Integral Evaluation – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.