Evaluating a Definite Integral In Exercises 9–36, evaluate the definite

Berikut ini adalah pertanyaan dari adit3295 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Evaluating a Definite Integral In Exercises 9–36, evaluate the definite integral. Use a graphing utility to verify your result.Bantuin Nomor 30 & 34 donkkk

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

$\begin{aligned}\sf30.\ &\int_{0}^{\pi}(2+\cos x)\,dx=\boxed{\,2\pi\,}\\\sf34.\ &\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx=\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}$
 

Penjelasan dengan langkah-langkah:

Integral Tentu (Definite Integral)

Nomor 30

$\begin{aligned}&\int_{0}^{\pi}(2+\cos x)\,dx\\&{=\ }\int_{0}^{\pi}2\,dx+\int_{0}^{\pi}\cos x\,dx\\&{=\ }\Bigl[2x\Bigr]_{0}^{\pi}+\Bigl[\sin x\Bigr]_{0}^{\pi}\\&{=\ }(2\pi-0)+(\sin\pi-\sin0)\\&{=\ }2\pi+(0-0)\\&{=\ }\boxed{\,2\pi\,}\end{aligned}$

Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 1.

Nilai integral: c = 6.2832.
Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):
2π = 2 × 3.1416 = 6.2832

⇒ Benar!
$\blacksquare$

Nomor 34

$\begin{aligned}&\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx\\&{=\ }\int_{\pi/4}^{\pi/2}2\,dx-\int_{\pi/4}^{\pi/2}\csc^2 x\,dx\\&{=\ }2\int_{\pi/4}^{\pi/2}dx+\int_{\pi/4}^{\pi/2}\left(-\csc^2 x\right)dx\\&{=\ }2\Bigl[x\Bigr]_{\pi/4}^{\pi/2}+\Bigl[\cot x\Bigr]_{\pi/4}^{\pi/2}\\&{=\ }2\left(\frac{\pi}{2}-\frac{\pi}{4}\right)+\left[\cot\left(\frac{\pi}{2}\right)-\cot\left(\frac{\pi}{4}\right)\right]\\&{=\ }2\cdot\frac{\pi}{4}+\left(0-1\right)\\&{=\ }\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}$

Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 2.

Nilai integral: c = 0.5708.
Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):
π/2 – 1 = 3.1416 – 1 = 1.5708 – 1 = 0.5708.

⇒ Benar!
$\blacksquare$

$[tex]\begin{aligned}\sf30.\ &\int_{0}^{\pi}(2+\cos x)\,dx=\boxed{\,2\pi\,}\\\sf34.\ &\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx=\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex] Penjelasan dengan langkah-langkah:Integral Tentu (Definite Integral)Nomor 30[tex]\begin{aligned}&\int_{0}^{\pi}(2+\cos x)\,dx\\&{=\ }\int_{0}^{\pi}2\,dx+\int_{0}^{\pi}\cos x\,dx\\&{=\ }\Bigl[2x\Bigr]_{0}^{\pi}+\Bigl[\sin x\Bigr]_{0}^{\pi}\\&{=\ }(2\pi-0)+(\sin\pi-\sin0)\\&{=\ }2\pi+(0-0)\\&{=\ }\boxed{\,2\pi\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 1.Nilai integral: c = 6.2832.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):2π = 2 × 3.1416 = 6.2832⇒ Benar![tex]\blacksquare[/tex]Nomor 34[tex]\begin{aligned}&\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx\\&{=\ }\int_{\pi/4}^{\pi/2}2\,dx-\int_{\pi/4}^{\pi/2}\csc^2 x\,dx\\&{=\ }2\int_{\pi/4}^{\pi/2}dx+\int_{\pi/4}^{\pi/2}\left(-\csc^2 x\right)dx\\&{=\ }2\Bigl[x\Bigr]_{\pi/4}^{\pi/2}+\Bigl[\cot x\Bigr]_{\pi/4}^{\pi/2}\\&{=\ }2\left(\frac{\pi}{2}-\frac{\pi}{4}\right)+\left[\cot\left(\frac{\pi}{2}\right)-\cot\left(\frac{\pi}{4}\right)\right]\\&{=\ }2\cdot\frac{\pi}{4}+\left(0-1\right)\\&{=\ }\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 2.Nilai integral: c = 0.5708.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):π/2 – 1 = 3.1416 – 1 = 1.5708 – 1 = 0.5708.⇒ Benar![tex]\blacksquare[/tex]$ $[tex]\begin{aligned}\sf30.\ &\int_{0}^{\pi}(2+\cos x)\,dx=\boxed{\,2\pi\,}\\\sf34.\ &\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx=\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex] Penjelasan dengan langkah-langkah:Integral Tentu (Definite Integral)Nomor 30[tex]\begin{aligned}&\int_{0}^{\pi}(2+\cos x)\,dx\\&{=\ }\int_{0}^{\pi}2\,dx+\int_{0}^{\pi}\cos x\,dx\\&{=\ }\Bigl[2x\Bigr]_{0}^{\pi}+\Bigl[\sin x\Bigr]_{0}^{\pi}\\&{=\ }(2\pi-0)+(\sin\pi-\sin0)\\&{=\ }2\pi+(0-0)\\&{=\ }\boxed{\,2\pi\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 1.Nilai integral: c = 6.2832.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):2π = 2 × 3.1416 = 6.2832⇒ Benar![tex]\blacksquare[/tex]Nomor 34[tex]\begin{aligned}&\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx\\&{=\ }\int_{\pi/4}^{\pi/2}2\,dx-\int_{\pi/4}^{\pi/2}\csc^2 x\,dx\\&{=\ }2\int_{\pi/4}^{\pi/2}dx+\int_{\pi/4}^{\pi/2}\left(-\csc^2 x\right)dx\\&{=\ }2\Bigl[x\Bigr]_{\pi/4}^{\pi/2}+\Bigl[\cot x\Bigr]_{\pi/4}^{\pi/2}\\&{=\ }2\left(\frac{\pi}{2}-\frac{\pi}{4}\right)+\left[\cot\left(\frac{\pi}{2}\right)-\cot\left(\frac{\pi}{4}\right)\right]\\&{=\ }2\cdot\frac{\pi}{4}+\left(0-1\right)\\&{=\ }\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 2.Nilai integral: c = 0.5708.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):π/2 – 1 = 3.1416 – 1 = 1.5708 – 1 = 0.5708.⇒ Benar![tex]\blacksquare[/tex]$ $[tex]\begin{aligned}\sf30.\ &\int_{0}^{\pi}(2+\cos x)\,dx=\boxed{\,2\pi\,}\\\sf34.\ &\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx=\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex] Penjelasan dengan langkah-langkah:Integral Tentu (Definite Integral)Nomor 30[tex]\begin{aligned}&\int_{0}^{\pi}(2+\cos x)\,dx\\&{=\ }\int_{0}^{\pi}2\,dx+\int_{0}^{\pi}\cos x\,dx\\&{=\ }\Bigl[2x\Bigr]_{0}^{\pi}+\Bigl[\sin x\Bigr]_{0}^{\pi}\\&{=\ }(2\pi-0)+(\sin\pi-\sin0)\\&{=\ }2\pi+(0-0)\\&{=\ }\boxed{\,2\pi\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 1.Nilai integral: c = 6.2832.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):2π = 2 × 3.1416 = 6.2832⇒ Benar![tex]\blacksquare[/tex]Nomor 34[tex]\begin{aligned}&\int_{\pi/4}^{\pi/2}(2-\csc^2 x)\,dx\\&{=\ }\int_{\pi/4}^{\pi/2}2\,dx-\int_{\pi/4}^{\pi/2}\csc^2 x\,dx\\&{=\ }2\int_{\pi/4}^{\pi/2}dx+\int_{\pi/4}^{\pi/2}\left(-\csc^2 x\right)dx\\&{=\ }2\Bigl[x\Bigr]_{\pi/4}^{\pi/2}+\Bigl[\cot x\Bigr]_{\pi/4}^{\pi/2}\\&{=\ }2\left(\frac{\pi}{2}-\frac{\pi}{4}\right)+\left[\cot\left(\frac{\pi}{2}\right)-\cot\left(\frac{\pi}{4}\right)\right]\\&{=\ }2\cdot\frac{\pi}{4}+\left(0-1\right)\\&{=\ }\boxed{\,\frac{\pi}{2}-1\,}\end{aligned}[/tex]Dengan sebuah graphing utility, diperoleh hasil seperti pada gambar 2.Nilai integral: c = 0.5708.Dengan π = 3.1416 (pendekatan 4 angka desimal di belakang koma):π/2 – 1 = 3.1416 – 1 = 1.5708 – 1 = 0.5708.⇒ Benar![tex]\blacksquare[/tex]$

Semoga dengan pertanyaan yang sudah terjawab oleh henriyulianto dapat membantu memudahkan mengerjakan soal, tugas dan PR sekolah kalian.

Apabila terdapat kesalahan dalam mengerjakan soal, silahkan koreksi jawaban dengan mengirimkan email ke yomemimo.com melalui halaman Contact

Last Update: Wed, 18 Jan 23

<< Previous Post

Next Post >>

Kerjakan soal, tugas, & PR sekolah semakin mudah

Evaluating a Definite Integral In Exercises 9–36, evaluate the definite

Jawaban dan Penjelasan

Penjelasan dengan langkah-langkah:

Video Tutorial Android dan Smartphone

Pertanyaan Lain Mata pelajaran Matematika