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Consider the curve segments: S1: y = x² from x == to x = 3 and 52: y = √√x from x = to x = 9. Set up integrals that give the arc lengths of the curve segments by integrating with respect to x. Demonstrate a substitution that verifies that these two integrals are equal. Substitution = √√x made in the integral L₂ = j√ ₁ + 1 = ² dx verifies that the length of the second segment is equal to 2x the length of the first segment: L₁ = 4 = √ √2x + 1dx. Substitution = x² made in the integral L₂ = 1+dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √√4x²+ Substitution = x² made in the integral L₂ = 1+1=dx dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = 4 - / Substitution u = 3x² made in the integral L₂ = ・i √ ₁ + 1 = ² verifies that the length of the second segment is equal to 4x the length of the first segment: L., = √4x² + 1dx. + Substitution" = √x made in the integral L₂ = -- / √ ₁ + 1 = dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = - / --/V₁-T -- / Vart + Tile - / √ √4x²+1dx. √4x²+1dx.

Consider the curve segments: S1: y = x² from x == to x = 3 and 52: y = √√x from x = to x = 9. Set up integrals that give the arc lengths of the curve segments by integrating with respect to x. Demonstrate a substitution that verifies that these two integrals are equal. Substitution = √√x made in the integral L₂ = j√ ₁ + 1 = ² dx verifies that the length of the second segment is equal to 2x the length of the first segment: L₁ = 4 = √ √2x + 1dx. Substitution = x² made in the integral L₂ = 1+dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √√4x²+ Substitution = x² made in the integral L₂ = 1+1=dx dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = 4 - / Substitution u = 3x² made in the integral L₂ = ・i √ ₁ + 1 = ² verifies that the length of the second segment is equal to 4x the length of the first segment: L., = √4x² + 1dx. + Substitution" = √x made in the integral L₂ = -- / √ ₁ + 1 = dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = - / --/V₁-T -- / Vart + Tile - / √ √4x²+1dx. √4x²+1dx.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the curve segments: S1: y = x² from x = to x = 3 and 1 S2: y = √√√x from x = - to x = 9. Set up integrals that give the arc lengths of the curve segments by integrating with respect to x. Demonstrate a substitution that verifies that these two integrals are equal. Substitutionu = √x made in the integral L2 = =/√₁+ 1+ dx verifies that the length of the second segment is equal to 2x the length of the first segment: L₁ = +-/Vix+Talk. √√2x + 1dx. 9 1 Substitution u = x² made in the integral L₂ = = 1+ -dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √ √4x² + 1dx. 9 Substitution u = x² made in the integral L₂ = -/v j√ITI 1+ dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = + = / + 1dx. Substitutionu = 3x² made in the integral L = - /√₁+I 1 + dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √ √4x² + 1dx. = / V4P² = Tak 9 Substitutionu = √x made in the integral L - / √ + Lave tes that = 1 + dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = - / √4x²+ √4x² + 1dx.
Transcribed Image Text:Consider the curve segments: S1: y = x² from x = to x = 3 and 1 S2: y = √√√x from x = - to x = 9. Set up integrals that give the arc lengths of the curve segments by integrating with respect to x. Demonstrate a substitution that verifies that these two integrals are equal. Substitutionu = √x made in the integral L2 = =/√₁+ 1+ dx verifies that the length of the second segment is equal to 2x the length of the first segment: L₁ = +-/Vix+Talk. √√2x + 1dx. 9 1 Substitution u = x² made in the integral L₂ = = 1+ -dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √ √4x² + 1dx. 9 Substitution u = x² made in the integral L₂ = -/v j√ITI 1+ dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = + = / + 1dx. Substitutionu = 3x² made in the integral L = - /√₁+I 1 + dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = √ √4x² + 1dx. = / V4P² = Tak 9 Substitutionu = √x made in the integral L - / √ + Lave tes that = 1 + dx verifies that the length of the second segment is equal to 4x the length of the first segment: L₁ = - / √4x²+ √4x² + 1dx.
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