Calculate the integral ∫(2 to 3) (x^3) dx. (2023)

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Question: Calculate the integral ∫(2 to 3) (x^3) dx. (2023)

Options:

Correct Answer: 8

Exam Year: 2023

Solution:

∫(2 to 3) (x^3) dx = [x^4/4] from 2 to 3 = (81/4 - 16/4) = 65/4 = 16.25.

Calculate the integral ∫(2 to 3) (x^3) dx. (2023)

Calculate the integral ∫(2 to 3) (x^3) dx. (2023)

Step 1: Identify the integral you need to calculate: ∫(2 to 3) (x^3) dx.
Step 2: Find the antiderivative of x^3. The antiderivative is x^4/4.
Step 3: Write the expression for the definite integral using the antiderivative: [x^4/4] from 2 to 3.
Step 4: Calculate the value of the antiderivative at the upper limit (3): (3^4)/4 = 81/4.
Step 5: Calculate the value of the antiderivative at the lower limit (2): (2^4)/4 = 16/4.
Step 6: Subtract the lower limit value from the upper limit value: (81/4) - (16/4) = (81 - 16)/4 = 65/4.
Step 7: Convert 65/4 to a decimal if needed: 65/4 = 16.25.

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 basic skill in calculus.