Q. 12 Set up and solve definite integr... [FREE SOLUTION]

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Chapter 6: Q. 12 (page 574)

Set up and solve definite integrals to find each volume, surface area, or arc length that follows. Solve each volume problem both with disks/washers and with shells, if possible.
The arc length of the curve is traced out by the graph of $f (x) = \ln (c s c x)$ on the interval $[\frac{蟺}{4}, \frac{蟺}{2}]$ .

Short Answer

Expert verified

$\ln (\sqrt{2} + 1)$ .

Step by step solution

Step 1. Given Information.

The volume of a given solid

$f (x) = \ln (c s c x)$ .

Step 2. Formulation.

The length of arc of a curve is

${鈭�}_{a}^{b} \sqrt{1 + y'^{2}} d x$ .

So, The arc is

$y = \ln (c s c x) y' = \frac{1}{c s c x} 路 (- c s c x c o t x) = - c o t x L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \sqrt{1 + {(- c o t x)}^{2}} d x L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \sqrt{1 + {(c o t x)}^{2}} d x [1 + c o t^{2} x = c s c^{2} x] L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \sqrt{c s c^{2} x} d x L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} c s c x d x L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \frac{c s c x (c s c x + c o t x)}{(c s c x + c o t x)} d x L = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \frac{c s c^{2} x + (c s c x c o t x)}{(c s c x + c o t x)} d x w e k n o w t h a t c s c x + c o t x = u 鈬� - c s c x c o t x - c s c^{2} x d x = d u L = - {[\ln |c s c x + c o t x|]}_{\frac{蟺}{4}}^{\frac{蟺}{2}} L = - \ln (1 + 0) + \ln (\sqrt{2} + 1) L = \ln (\sqrt{2} + 1)$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulation.

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91影视

Set up and solve definite integrals to find each volume, surface area, or arc length that follows. Solve each volume problem both with disks/washers and with shells, if possible.The arc length of the curve is traced out by the graph of f(x)=ln(cscx)on the interval 蟺4,蟺2.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.