Verify that F(x) is an antiderivative of the integrand f(x) and use Fundamental Theorem to...

Question:

Verify that {eq}F(x) {/eq} is an antiderivative of the integrand {eq}f(x) {/eq} and use Fundamental Theorem to evaluate the definite integral.

$$\int _0^{\pi} \sin(x) \ dx, \ F(x) = -\cos(x) $$

Fundamental Theorem of Calculus

If we are able to construct the antiderivative for a function that's being integrated, we can use it to evaluate a definite integral. The formula for doing so is the following.

{eq}\int_a^b f(x) dx = F(b) - F(a) {/eq}

Answer and Explanation:

The antiderivative of an integrand can be used to evaluate the overall integral. As we've been given a possible antiderivative, let's verify it to see if we can use it in this problem.

{eq}F'(x) = \frac{d}{dx} -\cos x = -(-\sin x) = \sin x {/eq}


This is indeed the integrand of this definite integral, so we can use it in the Fundamental Theorem of Calculus in order to find the value of this integral.

{eq}\begin{align*} \int_0^{\pi} \sin x dx &= -\cos x|_0^{\pi}\\ &= (-\cos \pi) - (-\cos 0)\\ &= -(-1) - (-1)\\ &= 1 + 1\\ &= 2 \end{align*} {/eq}


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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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