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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around the sun. (If your calculator or computer algebra system evaluates definite integrals, use it. Otherwise, use Simpson’s Rule.)

To determine

To find: The distance traveled by the planet.

Explanation

Given:

  • Eccentricity is 0.206.
  • Minimum distance from the sun is 4.6×107km .

Calculation:

Calculate minimum distance at perihelion using the formula.

r=α(1e) (1)

Substitute 4.6×107km for r , and 0.206 for e in the equation (1).

4.6×107=α(10.206)α=4.6×10710.206=4.6×1070.794

Write the equation for Mercury Orbit.

r=α(1e2)1+ecosθ (2)

Differentiate the equation (2) with respect to θ .

r=α(1e2)1+ecosθdrdθ=[(α(1e2))×(0+esinθ)][(1+ecosθ)×0](1+ecosθ)2[uv=udvvduv2]=[α(1e2)esinθ]0(1+ecosθ)2=α(1e2)esinθ(1+ecosθ)2

Calculate the value of r2+(drdθ)2 .

r2+(drdθ)2=[α(1e2)1+ecosθ]2+[α(1e2)esinθ(1+ecosθ)2]2=[α2(1e2)2(1+ecosθ)2]+[α2(1e2)2e2(sinθ)2(1+ecosθ)4]=[(α2(1e2)2)((1+ecosθ)2)]+[α2(

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