Calculate the integral ∫ from 0 to π of sin(x) dx.

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Question: Calculate the integral ∫ from 0 to π of sin(x) dx.

Options:

Correct Answer: 2

Solution:

The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.

Calculate the integral ∫ from 0 to π of sin(x) dx.

Calculate the integral ∫ from 0 to π of sin(x) dx.

Step 1: Identify the integral you need to calculate, which is ∫ from 0 to π of sin(x) dx.
Step 2: Find the antiderivative of sin(x). The antiderivative is -cos(x).
Step 3: Evaluate the antiderivative at the upper limit (π). Calculate -cos(π). Since cos(π) = -1, -cos(π) = 1.
Step 4: Evaluate the antiderivative at the lower limit (0). Calculate -cos(0). Since cos(0) = 1, -cos(0) = -1.
Step 5: Subtract the value at the lower limit from the value at the upper limit: 1 - (-1) = 1 + 1 = 2.
Step 6: Conclude that the value of the integral ∫ from 0 to π of sin(x) dx is 2.

Definite Integral – The process of calculating the area under the curve of a function over a specified interval.
Trigonometric Functions – Understanding the properties and integrals of sine and cosine functions.
Fundamental Theorem of Calculus – Connecting differentiation and integration, allowing evaluation of definite integrals using antiderivatives.