convert the following point coordinates into polar coordinates E(3,√3)
1. convert the following point coordinates into polar coordinates E(3,√3)
~ TrigonometrY
the cartesian coordinates (x , y) = (3 , √3) will be changed into polar coordinates (r , α) by following this form :
[tex]r = \sqrt{ {x}^{2} + {y}^{2} } \\ r = \sqrt{ {3}^{2} + {( \sqrt{3} )}^{2} } \\ r = \sqrt{9 + 3} \\ r = 2 \sqrt{3} \\ \\ \alpha = arc \: \tan( \frac{y}{x} ) \\ \alpha = arc \: \tan( \frac{ \sqrt{3} }{3} ) \\ \alpha = 30°[/tex]
Hence , the polar coordinates of (3 , √3) is (2√3 , 30°)
2. Evaluating a Definite Integral Using a Geometric Formula In Exercises 23–32, sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a>0, r>0).Bantuin Kalkulus nomor 32 donkkkk
Luas daerah yang dievaluasi adalah ½πr². Hal ini berarti:
[tex]\begin{aligned}\boxed{\vphantom{\Bigg|}\,\int_{-r}^{r}\sqrt{r^2-x^2}\,dx=\frac{1}{2}\pi r^2\,}\end{aligned}[/tex]
Diberikan integral:
[tex]\begin{aligned}\int_{-r}^{r}\sqrt{r^2-x^2}\,dx\end{aligned}[/tex]
dengan [tex]r > 0[/tex], yang akan dievaluasi dengan membuat sketsa daerahnya dan menghitung luasnya dengan rumus geometri.
Membuat Sketsa Daerah Integral
Kita tahu bahwa [tex]x^2 + y^2 = r^2[/tex] adalah persamaan lingkaran yang berpusat pada titik pusat koordinat [tex](0, 0)[/tex] dengan jari-jari [tex]r[/tex].
Maka, berdasarkan persamaan lingkaran tersebut, dapat diperoleh:
[tex]\begin{aligned}&y^2=r^2-x^2\\&\Rightarrow y=\pm\sqrt{r^2-x^2}\end{aligned}[/tex]
Artinya:
grafik [tex]y=\sqrt{r^2-x^2}[/tex] berbentuk setengah lingkaran pada arah sumbu-y positif, dangrafik [tex]y=-\sqrt{r^2-x^2}[/tex] berbentuk setengah lingkaran pada arah sumbu-y negatif.Oleh karena itu, daerah yang dievaluasi integralnya, yaitu [tex]y=\sqrt{r^2-x^2}[/tex] pada interval [tex][-r,r][/tex] berbentuk setengah lingkaran, dengan pusat (0,0) dan panjang jari-jari sebesar [tex]r[/tex]. Titik-titik potong pada sumbu koordinat adalah [tex](-r, 0)[/tex], [tex](0, r)[/tex], dan [tex](r, 0)[/tex].
Sketsa daerah pada sistem koordinat Cartesius terdapat pada gambar.
Evaluasi Integral Tentu dengan Rumus Geometri
Berdasarkan sketsa daerah yang telah digambarkan, luas daerah yang dievaluasi adalah setengah kali luas lingkaran dengan jari-jari [tex]r[/tex], yaitu ½πr².
Oleh karena itu, nilai integral tentu yang dievaluasi dapat dinyatakan dengan:
[tex]\displaystyle\int_{-r}^{r}\sqrt{r^2-x^2}\,dx=\frac{1}{2}\pi r^2[/tex]
[tex]\blacksquare[/tex]
________________
Tambahan:
Kita juga dapat menghitung integral tersebut dengan integral substitusi sebagai berikut.
[tex]\begin{aligned}&\int_{-r}^{r}\sqrt{r^2-x^2}\,dx\\&\quad\textsf{Substitusi trigonometri:}\\&\qquad x=r\sin u\\&\qquad\Rightarrow dx=r\cos u\,du\\&\quad\textsf{Interval $u$:}\\&\qquad u=\arcsin\left(\frac{x}{r}\right)\\&\qquad \Rightarrow \left[\arcsin\left(\frac{-r}{r}\right),\ \arcsin\left(\frac{r}{r}\right)\right]\\&\qquad \Rightarrow \left[\arcsin(-1),\ \arcsin(1)\right]\\&\qquad \Rightarrow \left[-\frac{\pi}{2},\ \frac{\pi}{2}\right]\\\end{aligned}[/tex]
[tex]\begin{aligned}&{=\ }\int_{-\pi/2}^{\pi/2}r\cos u\sqrt{r^2-r^2\sin^2u}\,du\\&{=\ }\int_{-\pi/2}^{\pi/2}r\cos u\sqrt{r^2\left(1-\sin^2u\right)}\,du\\&{=\ }\int_{-\pi/2}^{\pi/2}r\cos u\sqrt{r^2\cos^2u}\,du\\&{=\ }\int_{-\pi/2}^{\pi/2}r\cos u\cdot r\cos u\,du\\&{=\ }\int_{-\pi/2}^{\pi/2}r^2\cos^2u\,du\\&{=\ }r^2\cdot\int_{-\pi/2}^{\pi/2}\cos^2u\,du\\&{=\ }r^2\cdot\int_{-\pi/2}^{\pi/2}\frac{1+\cos2u}{2}\,du\\&{=\ }\frac{1}{2}r^2\cdot\int_{-\pi/2}^{\pi/2}(1+\cos2u)\,du\end{aligned}[/tex]
[tex]\begin{aligned}&{=\ }\frac{1}{2}r^2\left[u+\frac{1}{2}\sin2u\right]_{-\pi/2}^{\pi/2}\\&{=\ }\frac{1}{2}r^2\left[\left(\frac{\pi}{2}+\frac{1}{2}\sin(\pi)\right)-\left(-\frac{\pi}{2}+\frac{1}{2}\sin(-\pi)\right)\right]\\&{=\ }\frac{1}{2}r^2\left[\left(\frac{\pi}{2}+0\right)-\left(-\frac{\pi}{2}+0\right)\right]\\&{=\ }\frac{1}{2}r^2\left[\frac{\pi}{2}+\frac{\pi}{2}\right]\\&{=\ }\frac{1}{2}r^2\cdot\pi\\&{=\ }\boxed{\,\frac{1}{2}\pi r^2\,}\\\end{aligned}[/tex]
[tex]\blacksquare[/tex]
3. . Evaluate the integral xe^4x²+1 dx
Integral dari [tex]\int x.e^{4x^2+1}~dx[/tex] adalah [tex]\frac{e^{4x^2+1}}{8} + C[/tex]Integral dari [tex]\int x.e^{4x^2}+1~dx[/tex] adalah [tex]\frac{e^{4x^2}}{8}+x + C[/tex]
.
PENDAHULUANIntegral adalah operasi invers atau operasi kebalikan dari turunan. Integral dapat memuat fungsi Aljabar dan fungsi trigonometri. Pada Integral Aljabar ,terbagi dalam 3 penyelesaian.
.
[tex]\textbf{Penyelesaian dengan cara Umum : } \\\int f(x) = \int ax^n~dx= \frac{a}{n+1}x^{n+1} \\\\\int f(x) = \int k(x^n)~dx = k \int x^n~dx= \frac{k}{n+1}x^{n+1} \\\\\int (f(x) \pm g(x))~dx = \int f(x)~dx \pm \int g(x)~dx \\\\[/tex]
.
[tex]\textbf{Penyelesaian dengan cara Subtitusi : } \\\int f(x).g(x)~dx = \int \frac{f(x)}{g'(x)}.u~du \\\\dimana : \\u : fungsi~g(x) \\f(x) : fungsi~berderajat~n \\g(x) : fungsi~berderajat~(n+1) \\\\[/tex]
.
[tex]\textbf{Penyelesaian dengan cara Parsial : } \\\int f(x).g(x)~dx \to \int u~dv = u.v - \int v~du \\\\dimana : \\u : fungsi~f(x)~atau~g(x)~yang~dapat~diturunkan~hingga~nol. \\dv : fungsi~f(x)~atau~g(x)~yang~tidak~memiliki~limitasi~turunan.[/tex]
Mari simak penyelesaian berikut!
.
PEMBAHASANDIKETAHUI
[tex]\int x.e^{4x^2+1}~dx[/tex][tex]\int x.e^{4x^2}+1~dx[/tex].
DITANYA
[tex]Nilai~integral~...~?[/tex]
.
JAWAB
1. PENYELESAIAN PERTAMA.
Integral tersebut dapat diselesaikan dengan cara Subtitusi.
[tex]Misal : \\4x^2 +1 = u~~~\to~~~du = 8x~dx \\\\\boxed{dx = \frac{du}{8x}}[/tex]
.
[tex]\int x.e^{4x^2+1}~dx = \int x.e^{u}~\frac{du}{8x} \\\\~~~~~~~~~~~~~~~~~~~ = \frac{1}{8} \int e^u~du \\\\~~~~~~~~~~~~~~~~~~~ = \frac{1}{8} e^u + C \\\\~~~~~~~~~~~~~~~~~~~ = \frac{1}{8}e^{4x^2+1} + C \\\\\boxed{\boxed{\int x.e^{4x^2+1}~dx = \frac{e^{4x^2+1}}{8} + C}}[/tex]
.
2. PENYELESAIAN KEDUA.
Integral tersebut dapat diselesaikan dengan cara Subtitusi.
[tex]Misal : \\4x^2 = u~~~\to~~~du = 8x~dx \\\\\boxed{dx = \frac{du}{8x}}[/tex]
.
[tex]\int x.e^{4x^2}+1~dx = \int x.e^{u}~\frac{du}{8x} +\int 1~dx\\\\~~~~~~~~~~~~~~~~~~~~~ = \frac{1}{8} \int e^u~du+x +c_2 \\\\~~~~~~~~~~~~~~~~~~~~~ = \frac{1}{8} (e^u)+c_1+x+c_2 \\\\~~~~~~~~~~~~~~~~~~~~~ = \frac{1}{8}(e^{4x^2})+x + C \\\\\boxed{\boxed{\int x.e^{4x^2}+1~dx = \frac{e^{4x^2}}{8}+x + C}}[/tex]
.
KESIMPULAN
Jadi, nilai integral tersebut adalah
[tex]Jika~ \int x.e^{4x^2+1}~dx = \frac{e^{4x^2+1}}{8} + C[/tex][tex]Jika~ \int x.e^{4x^2}+1~dx = \frac{e^{4x^2}}{8}+x + C[/tex].
Catatan : Karena penulisan fungsi integral bernilai ambigu, silahkan sesuaikan dengan Integral dari persoalan yang dimaksud.
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PELAJARI LEBIH LANJUTMateri Integral Parsial : brainly.co.id/tugas/28945863
Materi Integral Subtitusi : brainly.co.id/tugas/30095944
Materi Integral Trigonometri : brainly.co.id/tugas/28945391
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_______________________________________________
DETAIL JAWABANKelas : 11
Mapel : Matematika
Materi : Integral Tak Tentu Fungsi Aljabar
Kode Kategorisasi : 11.2.10
4. Evaluate the techniques and models used by Ptolemy and Copernicus to explain the setup of the universe.
Jawaban:
Ptolemy believed in the theory that the earth was the center of the universe (geocentric) and it is not moving in the center while Copernicus later proposed otherwise, in his heliocentric theory he believed that the sun was the center of the universe but it was not moving. Both theories were influenced by their observation of how the light shade from the sun changes during the day, and also through observation of other celestial bodies whose light intensity changes which made them arrived to the conclusion that they are not equidistant nor stationary with respect to earth.
5. Complete the sentence by changing the verbs Yolanda (desiccate) ... some coconuts to make her coconut cake.
Yolanda desiccating some coconuts to make her coconut cake.
setahuku ( butuh pembenaran)
Yolanda desiccated some coconuts to make her coconut cake.
6. fill in the blanks by changing the verbs on the brackets to its suitable tense!
Jawaban:
1. was discussing
2. was
3. is moving
4. had tried
5. has explained
Jawaban:
1. Was discussing
2. Was
3. Is moving
4. Had tried
5. Has explained
Penjelasan:.
MAAF KLO SALAH
Jadikan jawaban terbaik ya
7. COMPLETE THE SENTENCES BY CHANGING THE CORRECT FORM OF THE VERB IN THE BRACKET. PAY ATTENTION TO THE TENSES ASKED
1. I have putted the book in the box.
2. I had slept before she called me.
3. By the time the show is over, Tania will have performed for an hour.
4. The class has been outside for recess.
5. Astri tried to hide her plate because she had broken it.
6. By the time I am 18, I will have had 18 dolls from my sister.
7. My sister has been yoga classes for a month.
8. I had read almost all of the page of this book before my brother torn it.
9. Do you think that she will have prepared enough material for the presentation?
10. I had gotten perfect score on every exam until this English exam.
8. contoh kalimat noun changing due to the suffixes
"Happiness" is something that you will never appreciate when it is around.
Happy (adjective) --> Happiness (Noun Suffixes)
i hope this can help you^^
9. fill the blanks by changing the word form in the bracket
Jawaban:
1. Drinking
2. Going
3. Singing
4. Watching
5. Shopping
6. Listening
7. Playing
8. Coming
9. Traveling
10. Writing
Penjelasan:
Semoga membantu.
10. complete the text by changing
Jawaban:
i usually come on time to school. however, this morning i woke up late. it was seven o'clock when I arrived at school. so I was scolded and punished. it was a bad experience of mine. i will never stay up anymore
Penjelasan:
rumus simple past tense:
(+) S + Verb 2 + O/C
11. Changing one letter the word road to sail
Jawaban:
road=read
sail=soil
Penjelasan:
12. fill in the blanks by changing the verb in the brackets to its suitable tanse!
Jawaban:
artinya=isi bagian yang kosong dengan mengubah kata kerja dalam tanda kurung menjadi tanse yang sesuai! saya
13. fill in the blanks by changing the clause on the bracket to suitable noun to fit in the sentence!
1. My purchased has been cancelled because of my overused credit card.
2. The children had free time at school due to the teachers and parents meeting.
3. Due to the village terrible flood, my grandmother had to stay in my house.
4. The minister agreed with the proposal because of the proposal's benefit for country.
5. Because of my generous parents, all of the children in our family have received the best in everything.
6. We were late to the meeting due to the heavy traffic.
Penjelasan:1. Transaksiku telah dibatalkan dikarenakan oleh kartu kredit yang telah mencapai batas pemakaian.
2. Anak-anak mempunyai waktu luang di sekolah dikarenakan rapat orang tua dan guru.
3. Dikarenakan banjir dahsyat di desa, nenekku harus tinggal di rumahku.
4. Dikarenakan kebaikkan hati orang tuaku, semua anak di keluarga kami mendapatkan yang terbaik dalam segala hal.
5. Kita terlambat ke rapat dikarenakan lalu lintas yang padat.
Penggunaan because of dan due to diikuti dengan noun atau noun phrase, bukan kalimat utuh.
Contoh:
I put my clothes under the roof because of the sky turned dark. (False)
I put my clothes under the roof because of the dark sky. (True)
Aku meletakkan baju-bajuku di bawah atap dikarenakan langit yang gelap.
Pelajari lebih lanjut materi tentang Because of - Due to pada brainly.co.id/tugas/5732990
#BelajarBersamaBrainly
14. 1. Complete the second sentence by changing the adjective from the first sentence to an abstract noun.
Jawaban:
b. strength
c. agility
d. determination
Penjelasan:
strength (noun) yang artinya kekuatan, strong (adjective: kata sifat) yg artinya kuat
agility (noun): kelincahan, agile (adjective): lincah
determination (noun): penentuan, determine: pasti
semoga membantu.
15. fill the blanks by changing the word from in the bracket
Jawaban:
||FILLTHEBLANKSBYCHANGING THEWORD FROMINTHEBRACKET||
The koala is sleeping the tree to get food.Mr. Raditya is talking in the seminar.The journalist is reporting the news from Uni Emirates.The birds are singingall day long.My brother is watchingfilmwith hus friends.We are makinga birthday cake for our father.My students are drawing animal at the moment.My mother is wateringthe plants every morning.The students are prayingafter class.The government is establishing a new office in Kalimantan.Penjelasan:
||PRESNETCONTINUOUSTENSE||
Kalimat positive (+)
→ S + to be (is/am/are) + verb-ing + O
Contoh :
We are studying English now.My mother is cooking in the kitchen at the moment.They are playing basketball.She is doing her homework right now.Kalimat negative (-)
→ S + to be (is/am/are) + not + verb-ing + O
Contoh :
We aren't studying English now.My mother isn't cooking in the kitchen at the moment.They aren't playing basketball.She isn't doing her homework right now.Kalimat nterrogative (?)
→ to be (is/am/are) + S + verb-ing + O
Contoh :
Arewe studying English now?Ismy mother cooking in the kitchen at the moment?Arethey playing basketball?Isshe doing her homework right now?Note:
Subject → to be
I → amHe, she, it → isThey, we, you → are16. rewrite the sentences by changing the bold print with suitable gerund!1.To smoke is dangerous to the health
Penjelasan:
yang di bold to smokw ya?? itu jadi smoking
17. fil in the blanks by changing the verb in the brackets to its suitable tense!TOLONG DI JAWAB KAK?
Jawaban:
5. Did
6. Cooking
7. Planning
8. Eating
9. Perfomed
10. Slept
18. fill in the blanks by changing the verb in the brackets to its suitable tanse!
Jawaban:
isi bagian yang kosong dengan mengubah kata kerja dalam tanda kurung menjadi tanse yang sesuai! saya
Penjelasan:
saya hanya jawab ya ng anda tulis dan bukan yg anda foto maaf jiga salah jawab nya ya
19. Fill in the blanks with the appropriate verb by changing it.1. John prefers (observe) ______ the moon to (finish) ________ the homework.
Jawaban:
John prefers observes the moon to finish the homework
John lebih suka mengamati bulan untuk menyelesaikan pekerjaan rumahnya
Maaf Kalo salah
20. Two points in the xy plane have Cartesian coordinates (2.00, 24.00) m and (23.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates.
Jawaban:
Dua titik pada bidang xy memiliki koordinat Kartesius (2.00, 24.00) m dan (23.00, 3.00) m. Tentukan (a) jarak antara titik-titik ini dan (b) kutubnya koordinat.
Penjelasan:
maaf kalau salah
kalau benar di follow ya gak maksa kok
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