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Q.16

Expert-verifiedFound in: Page 385

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [−π, π].

(b) The average value of f(x) = cos x on [0, 2π].

(c) The area of the region between the graphs of f(x) =$\sqrt{4-{x}^{2}}\text{and}g\left(x\right)=-\sqrt{4-{x}^{2}}\text{on}[-2,2]\text{.}$

Part (a) 0

Part (b) 0

Part (c) 12.56

The function is,$f\left(x\right)=\mathrm{cos}x$.

The objective is to find the signed area on $[-\pi ,\pi ]$without calculating.

Consider the following graph of the function:

From the graph it is seen that two areas are negative while two areas are positive of equal areas.

Therefore, the signed area is 0.

The function is,$f\left(x\right)=\mathrm{cos}x$

The objective is to find the average value on $[0,2\pi ]$without calculating.

From the above graph it is seen that both the positive area bis equal to the negative area.

Therefore, the average value is 0.

$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{4-{\mathrm{x}}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{g}\left(\mathrm{x}\right)=-\sqrt{4-{\mathrm{x}}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{objective}\mathrm{is}\mathrm{to}\mathrm{find}\mathrm{area}\mathrm{between}\mathrm{f}\left(\mathrm{x}\right)\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)\mathrm{on}[-2,2]$

$\mathrm{Consider}\mathrm{the}\mathrm{graph}:\phantom{\rule{0ex}{0ex}}\mathrm{It}\mathrm{is}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{radius}=4.\mathrm{Hence},\mathrm{area}\mathrm{is}:\phantom{\rule{0ex}{0ex}}\mathrm{Area}=\mathrm{\pi}{\left(2\right)}^{2}\mathrm{Area}=\mathrm{\pi}\left(4\right)\phantom{\rule{0ex}{0ex}}\mathrm{Area}=3.14\times 4\phantom{\rule{0ex}{0ex}}\mathrm{Area}=12.56$

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