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Unformatted text preview: 1 Mathematics 1c: Solutions, Homework Set 5 Due: Monday, May 10th at 10am. 1. (10 Points) Section 5.1, Exercise 4 Using Cavalieris 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 Cavalieris 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 xy = 2 F yx = f ( x,y ) . Use this example to discuss the relationship between Fubinis 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 yx ( x,y ) = f ( x,y ) . In the reverse order, we first apply Fubinis Theorem and then the Fundamen- tal Theorem of Calculus twice. Thus F ( x,y ) = Z y c Z x a f ( u,v ) dudv and...
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
- Spring '08