1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Tomtit [17]
3 years ago
10

Find the area of the region that lies inside the first curve and outside the second curve.

Mathematics
1 answer:
marishachu [46]3 years ago
7 0

Answer:

Step-by-step explanation:

From the given information:

r = 10 cos( θ)

r = 5

We are to find the  the area of the region that lies inside the first curve and outside the second curve.

The first thing we need to do is to determine the intersection of the points in these two curves.

To do that :

let equate the two parameters together

So;

10 cos( θ) = 5

cos( θ) = \dfrac{1}{2}

\theta = -\dfrac{\pi}{3}, \ \  \dfrac{\pi}{3}

Now, the area of the  region that lies inside the first curve and outside the second curve can be determined by finding the integral . i.e

A = \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} (10 \ cos \  \theta)^2 d \theta - \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \ \  5^2 d \theta

A = \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} 100 \ cos^2 \  \theta  d \theta - \dfrac{25}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \ \   d \theta

A = 50 \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \begin {pmatrix}  \dfrac{cos \ 2 \theta +1}{2}  \end {pmatrix} \ \ d \theta - \dfrac{25}{2}  \begin {bmatrix} \theta   \end {bmatrix}^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}}

A =\dfrac{ 50}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \begin {pmatrix}  {cos \ 2 \theta +1}  \end {pmatrix} \ \    d \theta - \dfrac{25}{2}  \begin {bmatrix}  \dfrac{\pi}{3} - (- \dfrac{\pi}{3} )\end {bmatrix}

A =25  \begin {bmatrix}  \dfrac{sin2 \theta }{2} + \theta \end {bmatrix}^{\dfrac{\pi}{3}}_{\dfrac{\pi}{3}}    \ \ - \dfrac{25}{2}  \begin {bmatrix}  \dfrac{2 \pi}{3} \end {bmatrix}

A =25  \begin {bmatrix}  \dfrac{sin (\dfrac{2 \pi}{3} )}{2}+\dfrac{\pi}{3} - \dfrac{ sin (\dfrac{-2\pi}{3}) }{2}-(-\dfrac{\pi}{3})  \end {bmatrix} - \dfrac{25 \pi}{3}

A = 25 \begin{bmatrix}   \dfrac{\dfrac{\sqrt{3}}{2} }{2} +\dfrac{\pi}{3} + \dfrac{\dfrac{\sqrt{3}}{2} }{2} +   \dfrac{\pi}{3}  \end {bmatrix}- \dfrac{ 25 \pi}{3}

A = 25 \begin{bmatrix}   \dfrac{\sqrt{3}}{2 } +\dfrac{2 \pi}{3}   \end {bmatrix}- \dfrac{ 25 \pi}{3}

A =    \dfrac{25 \sqrt{3}}{2 } +\dfrac{25 \pi}{3}

The diagrammatic expression showing the area of the region that lies inside the first curve and outside the second curve can be seen in the attached file below.

Download docx
You might be interested in
Write an inequality to represent the solution graphed below.
grin007 [14]

Answer:

x<-4

Step-by-step explanation:

open circle means 4 is not included

3 0
3 years ago
Read 2 more answers
Triangle ABC was dilated using the rule Do,4. Triangle
AysviL [449]

The value of OB based on the information given about the triangle is 3 units.

<h3>How to illustrate the information?</h3>

It should be noted that dilation is the process of enlarging or shrinking of an object.

In this case, the triangle is dilated. Therefore, the length of OB will be:

= 3/4 × 4 = 3 units.

Therefore, the correct option is B.

Learn more about triangle on:

brainly.com/question/16771919

#SPJ1

8 0
2 years ago
In how many ways can 5 different books be arranged on a shelf? 5 24 120
vlabodo [156]
120 ways. More details in attachment.

7 0
3 years ago
If a(x) = 2x - 4 and b(x) = x + 2, which of the following expressions produces a quadratic function?
pashok25 [27]

Answer: (an)(x)

Step-by-step explanation:

quizlet

8 0
3 years ago
Can someone help me with all of these
vazorg [7]
7. = 29/9 0.6 terminating
8. = 301/20 0.75 terminating

6 0
3 years ago
Other questions:
  • if you borrowed $1450 to buy a used car at a rate of 10% interest and planned on paying it back in 4 years, what is the total am
    6·2 answers
  • What percent of 12 is 9?
    11·1 answer
  • What is the value of x? <br> x=23 <br> x=35<br> x=58 <br> x=93
    14·2 answers
  • Translate six less than seven times a number is equal to -20 into an equation
    13·1 answer
  • Of the 29 houses on Jason's street, 3 are made out of brick. Approximately what percent of the houses are brick? ​
    15·1 answer
  • Math world problems, i think inequalities.
    8·1 answer
  • I need help............
    7·1 answer
  • 3. Triangle ABC has angle measures as noted below. Please note this triangle is not drawn to scale, and the longest leg of the f
    5·1 answer
  • HELP DUE IN 10 MINS!
    11·1 answer
  • Evelyn bought 8. 4 gallons of gas for $23. 94. Which equation can be used to determine the price of each gallon of gas? 8. 4x =
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!