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


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.

$$\int _1^4 \frac{1}{2\sqrt x} \ dx, \ F(x) = \sqrt x $$

Evaluating Definite Integral Using Fundamental Theorem of Calculus

Given a definite integral, we calculate its value using the Fundamental Theorem of Calculus. This requires that we find the anti-derivative of the integrand function of the definite integral and then evaluate the integral using the anti-derivative. In this question we are given a function and asked to show that it is the anti-derivative of the integrand function. Once we do that we may directly apply the Fundamental Theorem of Calculus to the anti-derivative function and get the answer.

Answer and Explanation:

To verify that {eq}F(x)=\sqrt {x} {/eq} is the anti-derivative of {eq}\displaystyle f(x)=\frac {1}{2 \sqrt {x}}, {/eq}

we show that {eq}F'(x)=f(x). {/eq}

To that end we have that

{eq}\displaystyle F'(x) = \left( \sqrt {x} \right)' = \left( x^{1/2} \right)' = \frac 12 x^{1/2-1} = \frac 12 x^{-1/2} = \frac {1}{2 \sqrt {x}}=f(x) \qquad (1) {/eq}

From (1) above it is shown that F(x) is the anti-derivative of f(x).

Therefore, from the the Fundamental Theorem of Calculus,

{eq}\displaystyle \int_1^4 \frac {1}{2 \sqrt {x}} \; dx = \left[ \sqrt {x} \right]_1^4 = \sqrt {4} - \sqrt {1} = 2-1=1. {/eq}

The answer is 1.

Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2

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