Quiz (+50): - Universityku

Terdapat bangun ruang prisma
Yang dibatasi sin(x), cos(x)
dan tan(x). Buktikan jika volumenya
2√2 – √5 – 1 – ln(½(√5 – 1)) satuan vol.

Quiz (+50):

Quiz (+50):

Perhatikan gambar

garis pembelah oranye tegak lurus dengan sumbu x

cari titik A :

cos(x) = tan(x)

cos(x) = sin(x)/cos(x)

cos²(x) = sin(x)

1 -sin²(x) = sin(x)

sin²(x) + sin(x) -1 = 0

sin²(x) + sin(x) + ¼ = 5/4

(sin(x) + ½)² = 5/4

sin(x) + ½ = ½√5

sin(x) = ½(√5 -1)

x = arcsin(½(√5 -1))

untuk batas ujung kanan :

sin(x) = cos(x)

sin(x)/cos(x) = 1

tan(x) = 1

x = ¼π

luas belahan bagian kanan :

$sf L_1 = int^{frac{pi}{4}}_{arcsin(frac{sqrt{5} -1}{2})} sf cos(x) - sin(x) : dx$

$sf L_1 = [ sf sin(x) + cos(x) ]^{frac{pi}{4}}_{arcsin(frac{sqrt{5} -1}{2})}$

$sf L_1 = [ sf sin(frac{pi}{4}) + cos(frac{pi}{4}) ] -[ sin(arcsin(frac{sqrt{5} -1}{2}))+ cos(arcsin(frac{sqrt{5} -1}{2}))]$

$sf L_1 = [ sf frac{1}{sqrt{2}} +frac{1}{sqrt{2}} ] -[ frac{sqrt{5} -1}{2} + sqrt{1 -( frac{sqrt{5} -1}{2})^2}]$

$sf L_1 = [ sf frac{2}{sqrt{2}}] -[ frac{sqrt{5} -1}{2} + sqrt{1 -frac{6 -2sqrt{5}}{4}}]$

$sf L_1 = sf sqrt{2} -[ frac{sqrt{5} -1}{2} + sqrt{frac{2sqrt{5} -2}{4}}]$

$sf L_1 = sf sqrt{2} -[ frac{sqrt{5} -1}{2} + sqrt{frac{sqrt{5} -1}{2}}]$

$sf L_1 = sf frac{2 sqrt{2} - sqrt{5} + 1}{2} - sqrt{frac{sqrt{5} -1}{2}}$

luas belahan bagian kiri :

$sf L_2 = sf int^{arcsin(frac{sqrt{5} -1}{2})}_0 sf tan(x) -sin(x) : dx$

$sf L_2 = sf [ cos(x) - ln(cos(x)) ]^{arcsin(frac{sqrt{5} -1}{2})}_0$

$sf L_2 = sf [ cos(arcsin(frac{sqrt{5} -1}{2})) - ln(cos(arcsin(frac{sqrt{5} -1}{2}))) ] - [ cos(0) - ln(cos(0)) ]$

$sf L_2 = sf [ sqrt{frac{sqrt{5} -1}{2}} - ln(sqrt{frac{sqrt{5} -1}{2}}) ] - [ 1 - ln(1) ]$

$sf L_2 = sf [ sqrt{frac{sqrt{5} -1}{2}} - frac{1}{2} ln(frac{sqrt{5} -1}{2}) ] - [ 1 - 0 ]$

$sf L_2 = sf sqrt{frac{sqrt{5} -1}{2}} - frac{1}{2} ln(frac{sqrt{5} -1}{2}) - 1$

Luas alas total :

$sf = L_1 + L_2$

$sf = frac{2 sqrt{2} - sqrt{5} + 1}{2} - sqrt{frac{sqrt{5} -1}{2}} + sqrt{frac{sqrt{5} -1}{2}} - frac{1}{2} ln(frac{sqrt{5} -1}{2}) - 1$

$sf = frac{2 sqrt{2} - sqrt{5} - 1}{2} - frac{1}{2} ln(frac{sqrt{5} -1}{2}) : satuan : luas$

Volume prisma :

= Luas alas × tinggi

$sf = ( frac{2 sqrt{2} - sqrt{5} - 1}{2} - frac{1}{2} ln(frac{sqrt{5} -1}{2})) times 2$

$sf = (2 sqrt{2} - sqrt{5} - 1 - ln(frac{sqrt{5} -1}{2}) ) : satuan : volume \ huge text{Terbukti}$

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