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
Sladkaya [172]
2 years ago
15

Find the area that the curve encloses and then sketch it. r = 3 + 8 sin(6)

Mathematics
1 answer:
Rudiy272 years ago
7 0

Answer:

A=41\pi\: \text{units}^2\approxA\approx128.8053\:\text{units}^2

Step-by-step explanation:

I assume you mean r=3+8\sin\theta:

Use the formula \displaystyle A=\int\limits^a_b \frac{1}{2} {r(\theta)^2} \, d\theta where a and b are the lower and upper bounds and r(\theta) is the equation of the polar curve.

Since the graph is symmetrical about the line \displaystyle \theta=\frac{\pi}{2}, let the bounds of integration be \displaystyle \biggr(-\frac{\pi}{2},\frac{\pi}{2}\biggr) to find half the area of the curve, and then find twice of that area:

\displaystyle A=\int\limits^a_b \frac{1}{2} {r(\theta)^2} \, d\theta\\\\A=2\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \frac{1}{2} {(3+8\sin\theta)^2} \, d\theta\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} 9+48\sin\theta+64\sin^2\theta \, d\theta\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} 9+48\sin\theta+64\biggr(\frac{1-\cos2\theta}{2} \biggr) \, d\theta\\\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} (9+48\sin\theta+32-32\cos2\theta) \, d\theta

\displaystyle A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} (41+48\sin\theta-32\cos2\theta) \, d\theta\\\\A=41\theta-48\cos\theta-16\sin2\theta\biggr|^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\\\\

A=\biggr[41\biggr(\frac{\pi}{2}\biggr)-48\cos\biggr(\frac{\pi}{2}\biggr)-16\sin2\biggr(\frac{\pi}{2}\biggr)\biggr]-\biggr[41\biggr(-\frac{\pi}{2}\biggr)-48\cos\biggr(-\frac{\pi}{2}\biggr)-16\sin2\biggr(-\frac{\pi}{2}\biggr)\biggr]\\\\A=\biggr[\frac{41\pi}{2}-24\sqrt{2}\biggr]-\biggr[-\frac{41\pi}{2}+24\sqrt{2}\biggr]\\ \\A=41\pi\\\\A\approx128.8053

Thus, the area of the curve is 41π square units. See below for a graph of the curve and its shaded area.

You might be interested in
Movie tickets cose $8. You want to buy as many tickets as possible with $36. Write and solve an inequality . Tell how many ticke
IrinaVladis [17]
36/8=t. The maximum amount of tickets I can buy are 4.
3 0
3 years ago
Liang enlarged a photograph to 1 3/4 times it original size. The original photograph was 5 inches wide. What was the width of th
Alja [10]
8.75 inches
5 divided by 4 is 1.25
1.25 x 3 = 3.75
3.75 + 5 = 8.75
7 0
3 years ago
What is the reflection in the x-axis & what’s the reflection in line y=x?
marishachu [46]

Answer:

  a) (3, -4), (2, 4), (-5, -6)

  b) (4, 3), (-4, 2), (6, -5)

Step-by-step explanation:

a) Reflection in the x-axis negates the y-coordinate:

  (x, y) ⇒ (x, -y)

  (3, 4) ⇒ (3, -4)

  (2, -4) ⇒ (2, 4)

  (-5, 6) ⇒ (-5, -6)

__

b) Reflection in the line y=x swaps the x- and y-coordinates:

  (x, y) ⇒ (y, x)

  (3, 4) ⇒ (4, 3)

  (2, -4) ⇒ (-4, 2)

  (-5, 6) ⇒ (6, -5)

8 0
4 years ago
Which is a graph of x^2/25 + y^2/4= 1
AVprozaik [17]

Answer:

If your answer is multiple choice, try to look for a graph that matches these intercepts. I hope this helped you. Have a great rest of your day!

7 0
3 years ago
Which best describes bias, if any, from the sample surveyed? survey: favorite sports sample: randomly selected fans at a basebal
Ugo [173]

Answer:

c

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • What is the solution to……………………ASAP
    9·1 answer
  • The volume of a sphere is 2,240 yd^3. What will be the new volume in the dimensions are multiplied by 1/4?
    11·1 answer
  • Is 1/3 less than or more than 1/7
    8·2 answers
  • Connie made deposits of $2,000 at the beginning of each year for four years. the rate she earned is 5% annually. what is the val
    7·1 answer
  • In Aunt Melly’s attics there are spiders and ants, the total of 152 legs and 22 heads. How many spiders and how many ants?
    12·1 answer
  • 16 cars are parked if 1/4 of them are blue how many cars are blue
    7·2 answers
  • Find x if the distance between the points (-5,x) and (-1,3) is 4 root 2 unit.​
    6·1 answer
  • Question is in picture pls ASAP its due by 8 its 6 will give you brainliest :)))))))
    5·1 answer
  • 2 Simplify: 2 (5+ 3x) + (x + 10)​
    15·1 answer
  • Johnny made 8 benches in 2 hours. At this rate, how
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!