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Solutions5

# Solutions5 - 1 Mathematics 1c Solutions Homework Set 5 Due...

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1 Mathematics 1c: Solutions, Homework Set 5 Due: Monday, May 10th at 10am. 1. (10 Points) Section 5.1, Exercise 4 Using Cavalieri’s principle, compute the volume of the structure shown in Figure 5.1.11 of the textbook; each section is a rectangle of length 5 and width 3 . Solution. By Cavalieri’s principle the volume of the solid in Figure 5.1.11 is the same as that of a rectangular parallelepiped of dimensions 3 × 5 × 7 or (3)(5)(7) = 105. 2. (20 Points) Section 5.2, Exercise 8 Let f be continuous on R = [ a, b ] × [ c, d ] . For a < x < b, and c < y < d , define F ( x, y ) = Z x a Z y c f ( u, v ) dv du. Show that 2 F ∂x∂y = 2 F ∂y∂x = f ( x, y ) . Use this example to discuss the relationship between Fubini’s Theorem and the equality of mixed partial derivatives. Solution. By the Fundamental Theorem of Calculus ∂F ∂x ( x, y ) = Z y c f ( x, v ) dv and applying it once again, we have 2 F ∂y∂x ( x, y ) = f ( x, y ) . In the reverse order, we first apply Fubini’s Theorem and then the Fundamen- tal Theorem of Calculus twice. Thus F ( x, y ) = Z y c Z x a f ( u, v ) du dv and ∂F ∂y ( x, y ) = Z x a f ( u, y ) du and then 2 F ∂x∂y ( x, y ) = f ( x, y ) .

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Solutions5 - 1 Mathematics 1c Solutions Homework Set 5 Due...

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