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Use the definite integral to find the area between the x-axis and f(x) over the indicated...

Question:

Use the definite integral to find the area between the {eq}x {/eq}-axis and {eq}f(x) {/eq} over the indicated interval.

{eq}f(x) = 4 - x^2; [0, 4] {/eq}

Definite integral:


A definite integral is used to calculate the area under a given curve and x-axis. The definite integral is calculated by finding the integrated function and subtracting the lower bound function value from the upper bound value.

Answer and Explanation:

Given Information

The function is:

{eq}f\left( x \right) = 4 - {x^2} {/eq}

Over the range 0 to 4.

So the area under the curve is:

{eq}\begin{align*} A = \int_0^4 {f\left( x \right)dx} \\ = \int_0^4 {\left( {4 - {x^2}} \right)dx} \\ = \left[ {4x - \dfrac{{{x^2}}}{2}} \right]_0^4\\ = 16 - 8\\ = 8 \end{align*} {/eq}

So, the area under the curve is 8 sq. units.


Learn more about this topic:

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Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2
1.7K

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