{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

13.1 Double Integrals

# 13.1 Double Integrals -...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 13.1 Double Integrals over Rectangles  R2:  Motivation  ________________________________________________    fx a     1.    2.        3.      4.      5.    Def:  ∫ b a   b n f ( x ) dx = lim ∑ f ( x k ) Δx Δx → 0 k =1  where  Δx  is the  width of each subinterval and xk is some point in the kth  subinterval.  R3:  Motivation _______________________________________________    0.5 0.0 2.0 0.5 1.5 1.0 0.5 0.0 1.0 1.      2.        3.        4.        5.      1.5 2.0 2.5     Def:  The definite integral of f(x, y) on rectangle R (a ≤ x ≤ b,  m n im c ≤ y ≤ d) is  ΔlA → 0 ∑ ∑ f ( x i , y j ) ΔA   Where  ΔA  is  the area of  i =1 j −1 each subrectangle on the xy plane and (xi, yj) is some point in  the ijth rectangle.    Suppose we slice the solid parallel to the y‐axis    0.5 0.0 2.0 0.5 1.5 1.0 0.5 0.0 1.0 1.      2.      3          1.5 2.0 2.5   Notes about the definite integral of a function of 2 variables:  1.   ∫∫ f ( x, y )dA  is a number – the limit of a Riemann sum.  2.   ∫∫ f ( x, y )dA  is a volume under z = f(x, y) and over R only if  R R f(x, y) ≥ 0 on R.    bd 3.  On a rectangle    63 Ex. Evaluate                                    ∫∫ 01 ∫∫ ac f ( x, y ) dydx = ∫ 2 x + 3 y + 5 dydx   d c ∫ b a f ( x, y ) dxdy   3 ∫∫ 6 Ex. Evaluate  1 0 2 x + 3 y + 5 dxdy                           Ex.  Find the volume of the solid bounded by the elliptic  paraboloid z = 1 + (x ‐1)2 + 4y2, the planes x = 3 and y = 2 and  the coordinate planes.                            2 Do:  Evaluate          ∫∫ 3 −1 1 xy 2 dydx   ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online