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Find \int_0^5 f(x)\,dx, if f(x) = 3 if x < 3 x if x \geq 3

Question:

Find {eq}\int_0^5 f(x)\,dx {/eq}, if

{eq}f(x) = \begin{cases} 3 & \text{ if } x < 3 \\ x **** \text{ if } x \geq 3 \end{cases} {/eq}

Definite Integral:

The definite integral of a real-value function {eq}f(x) {/eq} over an interval {eq}[a, b] {/eq} is calculated as follows

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

where {eq}F(x) {/eq} is the antiderivative of {eq}f(x). {/eq}

Answer and Explanation:

The definite integral

{eq}\displaystyle \int_0^5 f(x)\,dx {/eq},

where

{eq}f(x) = \begin{cases} 3 & \text{ if } x < 3 \\ x & \text{ if } x \geq 3 \end{cases} {/eq}

is calculated as follows

{eq}\displaystyle \int_0^5 f(x)\,dx = \int_0^3 3 \,dx + \int_3^5 x\,dx \\ \displaystyle = 3(3) + \frac{1}{2}(x^2)_3^5 \\ \displaystyle = 9 + \frac{1}{2}(25-9) \\ \displaystyle = 17. {/eq}


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
1.8K

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