Question

Draw the shape of region D limited by the xy=1 curve, y=x and x=e lines. Calculate the mass of the plate with density σ(x,y)=xy placed on this region.

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Answer to a math question Draw the shape of region D limited by the xy=1 curve, y=x and x=e lines. Calculate the mass of the plate with density σ(x,y)=xy placed on this region.

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Brice
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111 Answers
Verilen eğrilerin kesiştiği noktaları bulmak için;
1. y=x doğrusu ile xy=1 eğrisini çözelim.
2. x=e doğrusu ile xy=1 eğrisini çözelim.

Noktaları bulduktan sonra, bu üç doğru arasında kalan bölgenin şeklini çizeceğiz.

1. y=x doğrusu ile xy=1 eğrisini çözelim:
y=x \quad ve \quad xy=1 eşitlerini birbirine eşitleyerek çözelim:
x\cdot x = 1 \Rightarrow x^2 = 1 \Rightarrow x = 1
x = 1 değerini y=x doğrusuna koyarsak, y=1 olur.
Bu durumda kesişim noktası (1, 1) olur.

2. x=e doğrusu ile xy=1 eğrisini çözelim:
x=e \quad ve \quad xy=1 eşitlerini birbirine eşitleyerek çözelim:
e\cdot y = 1
y = \frac{1}{e}

Bu durumda kesişim noktası (e, \frac{1}{e}) olur.

Şimdi bu üç noktayı birbirleriyle birleştirerek, verilen bölgenin şeklini çizelim:
- (1, 1) noktası y=x doğrusu ve xy=1 eğrisiyle,
- (e, \frac{1}{e}) noktası x=e doğrusu ve xy=1 eğrisiyle,
- (e, \frac{1}{e}) ve (1, 1) noktaları y=x doğrusu ile birleştirilerek oluşturulur.

Şimdi bu bölgenin alanını hesaplayacağız. Bölge, bir üçgen şeklinde olduğundan, alanını bulmak için taban uzunluğunu ve yüksekliği hesaplayacağız:
Taban uzunluğu: e - 1
Yükseklik: \frac{1}{e} - 1

Bu durumda, bölgenin alanı:
A = \frac{1}{2} \cdot \text{Taban uzunluğu} \cdot \text{Yükseklik} = \frac{1}{2} \cdot (e - 1) \cdot \left(\frac{1}{e} - 1 \right)

Şimdi bu bölge üzerine yerleştirilen \sigma(x,y)=xy yoğunluklu levhanın kütlesini hesaplayalım. Bölgenin alanını kullanarak bu işlemi gerçekleştireceğiz:
\text{Kütle} = \iint_D \sigma(x,y) \, dA = \iint_D xy \, dA
\text{Kütle} = \int_1^e \int_1^\frac{1}{x} xy \, dydx

Yukarıdaki denklemi çözerek kütleyi hesaplayın.

\textbf{Yanıt:}
\text{Bölgenin Alanı: } A = \frac{1}{2} \cdot (e - 1) \cdot \left(\frac{1}{e} - 1 \right)
\text{Levhanın Kütle: } \text{Kütle} = \int_1^e \int_1^\frac{1}{x} xy \, dydx

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