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
Help I've got 5min to figure it out 20 points for 1st actual answer and brainlyist​
VARVARA [1.3K]
Chronological order
Hoped this help
4 0
2 years ago
A rational function g is given below. Find all values of a for which ​g(a)=16. g(x)=(4)/(x+1)+(20)/(x^(2)+2x+1)
aliina [53]

Answer:

a = 2/3

Step-by-step explanation:

(4(x+1) + 20 )/(x+1) = 16

4x + 4 + 20 = 16x + 16

12x = 8

x = 2/3

3 0
3 years ago
3 consecutive integers for 273
a_sh-v [17]

Answer:

90, 91 and 92

Step-by-step explanation:

Given

Consecutive integers = 273

Required

Find the integers

The question seem to be incomplete. However, I'll assume we're dealing with sum.

Let the smallest integer be y.

So,

y + y + 1 + y + 2 = 273

Collect like terms.

y + y + y = 273 - 2 - 1

3y = 270

Divide both sides by 3

y = 90

Hence, the integers are 90, 91 and 92

8 0
2 years ago
What is the answer to this equation? -108=8x
Vera_Pavlovna [14]
-108=8x
Divide 8 on both sides
-13.5=x
The answer is...X=-13.5


PLEASE MARK ME AS THE BRAINLIEST
4 0
3 years ago
Which of the following groups of measurements will form a triangle?<br>Help please.​
erastovalidia [21]

Answer:

It's C

Step-by-step explanation:

A triangles mesurement has to add up to 180. C is the only one that does.

3 0
2 years ago
Read 2 more answers
Other questions:
  • Convert.
    9·1 answer
  • Joshua is going to mow lawns during his summer break. The graph in the figure shows a linear relationship between the number of
    6·1 answer
  • Anna is buying a car and needs to borrow $23,500 from a bank offering a monthly interest rate of 0.4%. If Anna makes monthly pay
    7·1 answer
  • Can someone please help me out?
    14·2 answers
  • WILL MARK BRAINLIST TO WHOEVER ANSWERS FIRST!!!!!!!1
    13·1 answer
  • Could someone help i’ll mark brainliest! please explain too
    12·1 answer
  • PLEASE HELP ITS MY TRIMESTER TEST MY MOM WILL KILL ME IF I FAIL! &lt;3 WILL MARK BRAINLIST!
    15·2 answers
  • A.100<br> B.34<br> C.112<br> D.29<br> E.90<br> F.58<br> G.68
    5·1 answer
  • Find the distance between X (-3,8) and Z (-5,1). Round to the nearest tenth if necessary.
    9·1 answer
  • If the radius of a circle is doubled, how does the area of the circle change?
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!