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1) Find the average value of the function f(x) = X on [0,20] a) 2 b) 0 c) -1 d) None 2) Find the...

Question:

1) Find the average value of the function {eq}f(x) = X {/eq} on {eq}[0,20] {/eq}
{eq}a) 2 \\b) 0 \\c) -1 \\d) None {/eq}
2) Find the total area bounded by the graph of {eq}y = \sec^2 x{/eq} and the {eq}x-{/eq} axis on {eq}[-\frac{\pi}{4},\frac{\pi}{4} ] {/eq} {eq}a) 4 \\b) 2 \\c) 0 \\d) none {/eq}

Applications of Integrals:

Both the parts in the given problem relate to the application of integrals. For calculating the average value of a function, the formula used is given by: {eq}\begin{align*} \frac{1}{b-a}\int_{a}^{b} f(x) \ dx \end{align*} {/eq}. Also area under the curve is given by: {eq}\begin{align*} \int_{a}^{b} (\mathrm{upper \ curve - lower \ curve}) \ dx \end{align*} {/eq}.

Answer and Explanation: 1


{eq}f(x) = X {/eq} on {eq}[0,20] {/eq}.

Average value of the function over the interval {eq}\left[a, b \right] {/eq} is given by:

{eq}\begin{align*} \ & f_{avg}= \frac{1}{b-a}\int_{a}^{b} f(x) \ dx \end{align*} {/eq}

{eq}\begin{align*} \ & = \frac{1}{20-0}\int_{0}^{20} x \ dx \\ \\ \ & = \frac{1}{20} \left[\frac{x^2}{2} \right]_{0}^{20} \\ \\ \ & = \frac{1}{20} \left[\frac{(20)^2}{2} - \frac{0}{2} \right] \\ \\ \ & = \frac{1}{20} \left[\frac{400}{2} - 0 \right] \\ \\ \ & = \frac{1}{20} \left[200 \right] \\ \\ \ & = 10 \end{align*} {/eq}

Answer: d


2) Find the total area bounded by the graph of {eq}y = \sec^2 x{/eq} and the {eq}x-{/eq} axis on {eq}[-\frac{\pi}{4},\frac{\pi}{4} ] {/eq}

Area bounded by the curves is given by:

{eq}\begin{align*} &= \int_{a}^{b} (\mathrm{upper \ curve - lower \ curve}) \ dx \\ \\ &=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}(\sec^2x- 0) \ dx \\ \\ &= \left[\tan x \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \\ \\ &=\left[\tan \frac{\pi}{4}- \tan \frac{-\pi}{4} \right] \\ \\ &= \left[1- (-1) \right] \\ \\ &= 2 \ \mathrm{sq \ units} \end{align*} {/eq}

Answer: b


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Basic Calculus: Rules & Formulas

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Chapter 3 / Lesson 6
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In this lesson, we'll look at formulas and rules for differentiation and integration, which will give us the tools to deal with the operations found in basic calculus. At the end of the lesson, you'll also have the chance to test your new knowledge with a short quiz.


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