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
Dmitry_Shevchenko [17]
3 years ago
15

Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant below the line y=5 and betw

een the circles x2+y2=100 and x2−10x+y2=0.
Mathematics
1 answer:
vfiekz [6]3 years ago
3 0

First, complete the square in the equation for the second circle to determine its center and radius:

<em>x</em> ² - 10<em>x</em> + <em>y</em> ² = 0

<em>x</em> ² - 10<em>x</em> + 25 + <em>y </em>² = 25

(<em>x</em> - 5)² + <em>y</em> ² = 5²

So the second circle is centered at (5, 0) with radius 5, while the first circle is centered at the origin with radius √100 = 10.

Now convert each equation into polar coordinates, using

<em>x</em> = <em>r</em> cos(<em>θ</em>)

<em>y</em> = <em>r</em> sin(<em>θ</em>)

Then

<em>x</em> ² + <em>y</em> ² = 100   →   <em>r </em>² = 100   →   <em>r</em> = 10

<em>x</em> ² - 10<em>x</em> + <em>y</em> ² = 0   →   <em>r </em>² - 10 <em>r</em> cos(<em>θ</em>) = 0   →   <em>r</em> = 10 cos(<em>θ</em>)

<em>y</em> = 5   →   <em>r</em> sin(<em>θ</em>) = 5   →   <em>r</em> = 5 csc(<em>θ</em>)

See the attached graphic for a plot of the circles and line as well as the bounded region between them. The second circle is tangent to the larger one at the point (10, 0), and is also tangent to <em>y</em> = 5 at the point (0, 5).

Split up the region at 3 angles <em>θ</em>₁, <em>θ</em>₂, and <em>θ</em>₃, which denote the angles <em>θ</em> at which the curves intersect. They are

<em>θ</em>₁ = 0 … … … by solving 10 = 10 cos(<em>θ</em>)

<em>θ</em>₂ = <em>π</em>/6 … … by solving 10 = 5 csc(<em>θ</em>)

<em>θ</em>₃ = 5<em>π</em>/6  … the second solution to 10 = 5 csc(<em>θ</em>)

Then the area of the region is given by a sum of integrals:

\displaystyle \frac12\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}\left(10^2-(10\cos(\theta))^2\right)\,\mathrm d\theta+\int_{\frac\pi6}^{\frac{5\pi}6}\left((5\csc(\theta))^2-(10\cos(\theta))^2\right)\,\mathrm d\theta\right)

=\displaystyle 50\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\} \sin^2(\theta)\,\mathrm d\theta+\frac12\int_{\frac\pi6}^{\frac{5\pi}6}\left(25\csc^2(\theta) - 100\cos^2(\theta)\right)\,\mathrm d\theta

To compute the integrals, use the following identities:

sin²(<em>θ</em>) = (1 - cos(2<em>θ</em>)) / 2

cos²(<em>θ</em>) = (1 + cos(2<em>θ</em>)) / 2

and recall that

d(cot(<em>θ</em>))/d<em>θ</em> = -csc²(<em>θ</em>)

You should end up with an area of

=\displaystyle25\left(\left\{\int_0^{\frac\pi6}+\int_{\frac{5\pi}6}^{2\pi}\right\}(1-\cos(2\theta))\,\mathrm d\theta-\int_{\frac\pi6}^{\frac{5\pi}6}(1+\cos(2\theta))\,\mathrm d\theta\right)+\frac{25}2\int_{\frac\pi6}^{\frac{5\pi}6}\csc^2(\theta)\,\mathrm d\theta

=\boxed{25\sqrt3+\dfrac{125\pi}3}

We can verify this geometrically:

• the area of the larger circle is 100<em>π</em>

• the area of the smaller circle is 25<em>π</em>

• the area of the circular segment, i.e. the part of the larger circle that is bounded below by the line <em>y</em> = 5, has area 100<em>π</em>/3 - 25√3

Hence the area of the region of interest is

100<em>π</em> - 25<em>π</em> - (100<em>π</em>/3 - 25√3) = 125<em>π</em>/3 + 25√3

as expected.

You might be interested in
How many elements must a set have if the number of proper subsets of the set is one half of the total number of subsets of the​
lora16 [44]
The set  X = { x , {x} }  would, I think,  fit the Bill
So its 2 elements
6 0
3 years ago
What is 42/8832 with remainder
mario62 [17]
8832 ÷ 42 = 210.2857

3 0
3 years ago
Hey I’m confused on this problem help.
PilotLPTM [1.2K]

the answer is b carmella

7 0
2 years ago
How to calculate the are of a shape
Ira Lisetskai [31]
This can vary, some shapes have different area formulas, here are some basic formulas I can give you:
Circle: A = \pir^{2}Triangle: A = \frac{h_{b} b}{2}Square: A = a^{2}Rectangle: A = L*W
6 0
3 years ago
If cats are mammals and so are dogs, then why is my grass yellow
ipn [44]

Answer:

well......it may be pi$$-

Step-by-step explanation:

5 0
3 years ago
Other questions:
  • The corner where the states of Utah, New Mexico, and Colorado meet is called the Four Corners.
    12·2 answers
  • Zoe keeps track of the miles per gallon her car gets per week. She has accumulated the following data:
    12·2 answers
  • One serving of chad's cereal is 1 1/4 ounces. How many servins are in 15 1/3 ounces box?
    9·1 answer
  • 1/4 times 1/4 equals how much?
    10·2 answers
  • Help plzzzzz!!!!! (brainliest)
    11·2 answers
  • Martha is making a quilt. She has 63 squares ready to sew together. She wants the quilt to be 9 rows long. How many squares will
    12·1 answer
  • Find the unit price of 8 mugs that cost $38.32 total
    7·2 answers
  • I NEED YOUR HELP FAST!!!!
    10·2 answers
  • Diana is walking at 5 miles per hour and Gustavo is walking at 6 miles per hour and started 1 mile behind Diana. Will Gustavo ev
    7·2 answers
  • Letter=number<br> 14.6=e-3.2
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!