Evaluate \int_{4}^{9} \sqrt{x} dx.

Question:

Evaluate {eq}\int_{4}^{9} \sqrt{x}\; dx {/eq}.

First Fundamental Theorem:

It states that, if {eq}\displaystyle{ f(x) }{/eq} is continuous on the closed interval {eq}\displaystyle{ [a,b] }{/eq} and {eq}\displaystyle{ F }{/eq} is the indefinite integral of {eq}\displaystyle{ f (x) }{/eq} on {eq}\displaystyle{ [a,b], }{/eq} then:

{eq}\displaystyle{ \int _ { a } ^ { b } f ( x ) d x = F ( b ) - F ( a ) }{/eq}

and call this the definite integral of {eq}\displaystyle{ f(x) }{/eq} from {eq}\displaystyle{ a }{/eq} to {eq}\displaystyle{ b. }{/eq} This theorem describes the relationship between differentiation and integration, which are inverse functions of one another.

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We have,

{eq}\displaystyle{ \int_{4}^{9} \sqrt{x}\; dx }{/eq}

Apply the first fundamental theorem of calculus as:

{eq}\large{ \begin{array} { l...

The Fundamental Theorem of Calculus

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Chapter 12 / Lesson 10
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The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.