Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.

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Question: Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.

Options:

Correct Answer: 2

Solution:

The integral evaluates to [x^3/3 - 4x] from 1 to 3 = (27/3 - 12) - (1/3 - 4) = 2.

Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.

Evaluate ∫ from 1 to 3 of (x^2 - 4) dx.

Step 1: Identify the integral to evaluate: ∫ from 1 to 3 of (x^2 - 4) dx.
Step 2: Find the antiderivative of the function (x^2 - 4). The antiderivative is (x^3/3 - 4x).
Step 3: Evaluate the antiderivative at the upper limit (x = 3). Calculate (3^3/3 - 4*3). This gives (27/3 - 12) = 9 - 12 = -3.
Step 4: Evaluate the antiderivative at the lower limit (x = 1). Calculate (1^3/3 - 4*1). This gives (1/3 - 4) = 1/3 - 12/3 = -11/3.
Step 5: Subtract the lower limit result from the upper limit result: (-3) - (-11/3). Convert -3 to a fraction: -3 = -9/3. Now calculate: (-9/3) - (-11/3) = (-9 + 11)/3 = 2/3.
Step 6: The final result of the integral is 2/3.

Definite Integral – 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.
Polynomial Integration – Involves finding the integral of polynomial functions, which is a common type of function in calculus.