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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.

$$\int _0^1 \sqrt x \ dx, \ F(x) = \frac{2}{3}x^{3/2} $$

Fundamental Theorem of Calculus:

The differentiation of any integration of any function which is an actual function. For example, take the derivative of integration of function:

{eq}\displaystyle \frac{d}{dy} \int_b^y g(t) dt = g(y). {/eq}

Next, we will see an applications:

1. Radical rule: {eq}\displaystyle \sqrt{b}=b^{\frac{1}{2}}. {/eq}

2. The power rule of integration: {eq}\displaystyle \int u^bdu=\frac{u^{b+1}}{b+1}, \quad b\ne -1. {/eq}

3. Move the constant out: {eq}\displaystyle \left(b\cdot f\right)'=b\cdot f'. {/eq}

4. The power rule of derivative: {eq}\displaystyle \frac{d}{du}\left(u^b\right)=b\cdot u^{b-1}. {/eq}

Answer and Explanation:

We have to solve the integration of $$\displaystyle I = \int _0^1 \sqrt x \ dx $$

Use the radical rule.

$$\displaystyle = \int _0^1x^{\frac{1}{2}}dx $$

Apply the power rule.

$$\displaystyle = \left[\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]^1_0 $$

Simplify:

$$\displaystyle = \left[\frac{2}{3}x^{\frac{3}{2}}\right]^1_0 $$

Compute the boundaries.

$$\displaystyle = \left[\frac{2}{3}1^{\frac{3}{2}}-0\right] $$

Simplify:

$$\displaystyle = \frac{2}{3}. $$

Now, solving for $$\displaystyle F(x) = \frac{2}{3}x^{3/2} $$

Take derivative both sides.

$$\displaystyle \frac{d}{dx}F(x) = \frac{d}{dx}\left(\frac{2}{3}x^{\frac{3}{2}}\right) $$

Move the constant out.

$$\displaystyle F'(x) = \frac{2}{3}\frac{d}{dx}\left(x^{\frac{3}{2}}\right) $$

Apply power rule of derivative.

$$\displaystyle F'(x) = \frac{2}{3}\cdot \frac{3}{2}x^{\frac{3}{2}-1} $$

Simplify:

$$\displaystyle F'(x) = \sqrt{x} $$

Which is equal to {eq}f(x). {/eq}


Learn more about this topic:

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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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