Find the value of the integral ∫(0 to 1) (3x^2)dx.

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Question: Find the value of the integral ∫(0 to 1) (3x^2)dx.

Options:

Correct Answer: 1

Solution:

The integral ∫(3x^2)dx from 0 to 1 = [x^3] from 0 to 1 = 1.

Find the value of the integral ∫(0 to 1) (3x^2)dx.

Find the value of the integral ∫(0 to 1) (3x^2)dx.

Step 1: Identify the integral you need to solve, which is ∫(3x^2)dx from 0 to 1.
Step 2: Find the antiderivative of 3x^2. The antiderivative is x^3 because when you differentiate x^3, you get 3x^2.
Step 3: Write the expression for the definite integral using the antiderivative: [x^3] from 0 to 1.
Step 4: Evaluate the antiderivative at the upper limit (1): (1)^3 = 1.
Step 5: Evaluate the antiderivative at the lower limit (0): (0)^3 = 0.
Step 6: Subtract the lower limit result from the upper limit result: 1 - 0 = 1.
Step 7: The value of the integral ∫(3x^2)dx from 0 to 1 is 1.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Power Rule of Integration – A method for integrating polynomial functions, where ∫x^n dx = (x^(n+1))/(n+1) + C.