Q. 4 Each of the integral expressions... [FREE SOLUTION]

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Chapter 13: Q. 4 (page 1025)

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral
$\frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} ({(3 \sin 胃)}^{2} - {(1 + \sin 胃)}^{2}) d 胃$

Short Answer

Expert verified

The value of integral is $蟺$ square units.

Step by step solution

Step 1. Given information

Integral:

$\frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} ({(3 \sin 胃)}^{2} - {(1 + \sin 胃)}^{2}) d 胃$

Step 2. Plot the region

Since the area of a function $r = f (胃)$ is $\frac{1}{2} {鈭�}_{a}^{b} r^{2} d 胃$

So by comparing the given integral we get,

$r = 3 \sin 胃 r = 1 + \sin 胃$

So the graph of this curve is:

Step 3. Evaluate integral.

$\frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} ({(3 \sin 胃)}^{2} - {(1 + \sin 胃)}^{2}) d 胃 = \frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} (9 \sin^{2} 胃 - (1 + \sin^{2} 胃 + 2 \sin 胃)) d 胃 = \frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} (9 \sin^{2} 胃 - 1 - \sin^{2} 胃 - 2 \sin 胃) d 胃 = \frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} (8 \sin^{2} 胃 - 1 - 2 \sin 胃) d 胃 = \frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} (4 (1 - \cos 2 胃) - 1 - 2 \sin 胃) d 胃 = \frac{1}{2} {鈭�}_{\frac{蟺}{6}}^{\frac{5}{6}} (3 - 4 \cos 2 胃 - 2 \sin 胃) d 胃$

$= \frac{1}{2} {[3 胃 - \frac{4 s i n 2 胃}{2} + 2 \cos 胃]}_{\frac{蟺}{6}}^{\frac{5 蟺}{6}} = \frac{1}{2} [(3 脳 \frac{5 蟺}{6} - 2 \sin (\frac{5 蟺}{3}) + 2 \cos (\frac{5 蟺}{6})) - (3 脳 \frac{蟺}{6} - 2 \sin (\frac{蟺}{3}) + 2 \cos (\frac{蟺}{6}))] = \frac{1}{2} [\frac{5 蟺}{2} + \frac{2 \sqrt{3}}{2} - \frac{2 \sqrt{3}}{2} - \frac{蟺}{2} + 2 脳 \frac{\sqrt{3}}{2} - 2 脳 \frac{\sqrt{3}}{2}] = \frac{1}{2} (\frac{4 蟺}{2}) = 蟺$

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Plot the region

Step 3. Evaluate integral.

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91影视

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral 12鈭�蟺6563sin胃2-1+sin胃2d胃

Short Answer

Step by step solution

Step 1. Given information

Step 2. Plot the region

Step 3. Evaluate integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Calculus

Pure Maths

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.