Math Problem Statement

Evaluate the definite integral \( \int_{0}^{\pi} 2 \sin(x) \, dx \).

Solution

To evaluate the definite integral 0π2sin(x)dx\int_{0}^{\pi} 2 \sin(x) \, dx, follow these steps:

  1. Find the antiderivative: The antiderivative of 2sin(x)2 \sin(x) is 2cos(x)-2 \cos(x).

  2. Evaluate the definite integral using the limits: 2cos(x)0π=2cos(π)(2cos(0))\left. -2 \cos(x) \right|_{0}^{\pi} = -2 \cos(\pi) - (-2 \cos(0))

  3. Calculate the values:

    • cos(π)=1\cos(\pi) = -1
    • cos(0)=1\cos(0) = 1

    Substituting these values in: 2(1)(2×1)=2(2)=2+2=4-2(-1) - (-2 \times 1) = 2 - (-2) = 2 + 2 = 4

So, the value of the definite integral is 44.

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Math Problem Analysis

Mathematical Concepts

Definite Integrals
Trigonometric Functions

Formulas

The antiderivative of \( \sin(x) \) is \( -\cos(x) \)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12