I tested CalcMentor with this "integrate sin(x) from 0 to 2pi" What you say?
**Problem:** integrate sin(x) from 0 to 2pi
**Answer:** $$0$$
**Step 1: Recall the Antiderivative of Sine**
The antiderivative of $\sin(x)$ is $-\cos(x)$, because the derivative of $-\cos(x)$ is $\sin(x)$. This is a basic integration rule that we will apply.
**Step 2: Apply the Fundamental Theorem of Calculus**
The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ can be calculated as $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. For our case, $f(x) = \sin(x)$, $F(x) = -\cos(x)$, $a = 0$, and $b = 2\pi$.
**Step 3: Evaluate the Antiderivative at the Limits**
We evaluate $-\cos(x)$ at $x = 2\pi$ and $x = 0$. Since $\cos(2\pi) = \cos(0) = 1$, we have $-\cos(2\pi) - (-\cos(0)) = -1 - (-1)$.
**Step 4: Simplify the Expression**
Simplifying $-1 - (-1)$ gives us $0$. This means the area under the curve of $\sin(x)$ from $0$ to $2\pi$ is $0$.
Therefore, the answer is:
$$0$$
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*Solved by [CalcMentor AI](https://derivativecalculus.com/calcmentor.html) — Free · No Sign-up*
**Answer:** $$0$$
**Step 1: Recall the Antiderivative of Sine**
The antiderivative of $\sin(x)$ is $-\cos(x)$, because the derivative of $-\cos(x)$ is $\sin(x)$. This is a basic integration rule that we will apply.
**Step 2: Apply the Fundamental Theorem of Calculus**
The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ can be calculated as $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. For our case, $f(x) = \sin(x)$, $F(x) = -\cos(x)$, $a = 0$, and $b = 2\pi$.
**Step 3: Evaluate the Antiderivative at the Limits**
We evaluate $-\cos(x)$ at $x = 2\pi$ and $x = 0$. Since $\cos(2\pi) = \cos(0) = 1$, we have $-\cos(2\pi) - (-\cos(0)) = -1 - (-1)$.
**Step 4: Simplify the Expression**
Simplifying $-1 - (-1)$ gives us $0$. This means the area under the curve of $\sin(x)$ from $0$ to $2\pi$ is $0$.
Therefore, the answer is:
$$0$$
---
*Solved by [CalcMentor AI](https://derivativecalculus.com/calcmentor.html) — Free · No Sign-up*