Kenneth Kuttler

382 CHAPTER 20. THE INTEGRAL IN OTHER COORDINATES

PROCEDURE 20.5.3 Suppose U is an open subset of Rn for n > 0 and supposef : U → f (U) is a C1 function which is one to one, x= f (u). 2Then if h : f (U)→ R,∫

Uh(f (u))

∣∣∣∣ ∂ (x1, · · · ,xn)

∂ (u1, · · · ,un)

∣∣∣∣ dV =∫f(U)

h(x) dV.

Example 20.5.4 Find the area of the region in R2 which is determined by the lines y =2x,y = (1/2)x,x+ y = 1,x+ y = 3.

You might sketch this region. You will find it is an ugly quadrilateral. Let u = x+y andv = y

x . The reason for this is that the given region corresponds to (u,v) ∈ [1,3]×[ 1

2 ,2], a

nice rectangle. Now we need to solve for x,y to obtain the Jacobian. A little computationshows that

x =u

v+1, y =

uvv+1

Therefore, ∂ (x,y)∂ (u,v) is

det

 1v+1 − u

(v+1)2

vv+1

u(v+1)2

=u

(v+1)2 .

Therefore, the area of this quadrilateral is∫ 2

1/2

∫ 3

(v+1)2 dudv =43.

20.6 Exercises1. Verify the three dimensional volume increment in spherical coordinates is

ρ2 sin(φ)dρdφdθ .

2. Find the area of the bounded region R, determined by 5x+ y = 1,5x+ y = 9,y = 2x,and y = 5x.

3. Find the area of the bounded region R, determined by y+2x = 6,y+2x = 10,y = 3x,and y = 4x.

4. A solid, R is determined by 3x+ y = 2,3x+ y = 4,y = x, and y = 2x and the densityis ρ = x. Find the total mass of R.

5. A solid, R is determined by 4x+2y= 1,4x+2y= 9,y= x, and y= 6x and the densityis ρ = y. Find the total mass of R.

6. A solid, R is determined by 3x+y= 3,3x+y= 10,y= 3x, and y= 5x and the densityis ρ = y−1. Find the total mass of R.

2This will cause non overlapping infinitesimal boxes in U to be mapped to non overlapping infinitesimalparallelepipeds in V .

Also, in the context of the Riemann integral we should say more about the set U in any case the functionh. These conditions are mainly technical however, and since a mathematically respectable treatment will not beattempted for this theorem in this part of the book, I think it best to give a memorable version of it which isessentially correct in all examples of interest.

382 CHAPTER 20. THE INTEGRAL IN OTHER COORDINATESPROCEDURE 20.5.3 Suppose U is an open subset of R" for n > 0 and supposef :U > f (U) isaC! function which is one to one, x = f (u). *Then ifh: f (U) > R,[nse FaExample 20.5.4 Find the area of the region in R? which is determined by the lines y =2x,y = (1/2)x,x+y = 1xty=3.dV = | h(a) dV.#(U)You might sketch this region. You will find it is an ugly quadrilateral. Let u = x+y andv=. The reason for this is that the given region corresponds to (u,v) € [1,3] x [5.2].anice rectangle. Now we need to solve for x,y to obtain the Jacobian. A little computationshows thatu uvx= 5 =v+1 y v+l1Therefore d(xy) is> d(u,v)1 __u Zv+1 41 udet » ( ) = 5v+1 2 (v + 1)Therefore, the area of this quadrilateral is2 73 4Do [ pad = 5.1241 (v+1) 320.6 Exercises1. Verify the three dimensional volume increment in spherical coordinates isp’sin(¢)dpdodée.2. Find the area of the bounded region R, determined by 5x+ y= 1,5x+y =9,y = 2x,and y = 5x.3. Find the area of the bounded region R, determined by y+ 2x = 6,y+2x = 10,y = 3x,and y = 4x.4. A solid, R is determined by 3x+y = 2,3x+y =4,y =x, and y = 2x and the densityis p =x. Find the total mass of R.5. A solid, R is determined by 4x+ 2y = 1,4x+2y=9,y =x, and y = 6x and the densityis Pp = y. Find the total mass of R.6. A solid, R is determined by 3x+y = 3,3x+y= 10, y = 3x, and y = 5x and the densityis p = y_!. Find the total mass of R.This will cause non overlapping infinitesimal boxes in U to be mapped to non overlapping infinitesimalparallelepipeds in V.Also, in the context of the Riemann integral we should say more about the set U in any case the functionh. These conditions are mainly technical however, and since a mathematically respectable treatment will not beattempted for this theorem in this part of the book, I think it best to give a memorable version of it which isessentially correct in all examples of interest.