Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu

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Chabot Mathematics. §7.6 Double Integrals. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu. 7.5. Review §. Any QUESTIONS About §7.5 → Lagrange Multipliers Any QUESTIONS About HomeWork §7.5 → HW-8. Partial- Deriv↔Partial - Integ.

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Chabot Mathematics

§7.6 DoubleIntegrals

Bruce Mayer, PE

7.5

Review §
• §7.5 → Lagrange Multipliers
• §7.5 → HW-8
Partial-Deriv↔Partial-Integ
• Extend the Concept of a “Partial” Operation to Integration.
• Consider the mixed 2nd Partial
• ReWrite the Partial in Lebniz Notation:
• Now Let:
Partial-Deriv↔Partial-Integ
• Thus with ∂z/∂y = u:
• Now Multiply both sides by ∂x and Integrate
• Integration with respect to the Partial Differential, ∂x, implies that y is held CONSTANT during the AntiDerivation
Partial-Deriv↔Partial-Integ
• Performing The AntiDerivation while not including the Constant find:
• Now Let:
• Then substitute, then multiply by ∂x
Partial-Deriv↔Partial-Integ
• Integrating find:
• After AntiDerivation:
• But ReCall:
• Back Substituting find
• By the Associative Property
Partial-Deriv↔Partial-Integ
• Also ReCallClairaut’s Theorem:
• This Order-Independence also Applies to Partial Integrals Which leads to the Final Statement of the Double Integral
• C is the Constant of Integration
Area BETWEEN Curves
• As before Find Area by adding Vertical Strips.
• In this case for the Strip Shown:
• Width = Δx
• Height = ytop − ybot or
• Then the strip area
Area BETWEEN Curves
• Note that for every CONSTANT xk, that y runs:
• Now divide the Hgt into pieces Δy high
• So then ΔA:
• Then Astrip is simply the sum of all the small boxes
Area BETWEEN Curves
• Substitute:
• Then
• Next Add Up all the Strips to find the Total Area, A
Area BETWEEN Curves
• This Relation
• Is simply a Riemann Sum
• Then in the Limit
• Find
Example  Area Between Curves
• Find the area of the region contained between the parabolas
Example  Area Between Curves
• SOLUTION: Use Double Integration

% Bruce Mayer, PE

% MTH-16 • 22Feb14

% Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m

% Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E.

% Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN

% 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295

%

clc; clear; clf; % Clear Figure Window

%

% The Function

xmin = -2; xmax = 2;

ymin = 0; ymax = 10;

x = linspace(xmin,xmax,500);

y1 = -x.^2 + 9;

y2 = x.^2 + 1;

%

plot(x,y1,'--', x,y2,'m','LineWidth', 5), axis([0 xmaxyminymax]),...

grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = -x^2 + 9 & x^2 + 1'),...

title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...

legend('-x^2 + 9','x^2 + 1') %

display('Showing 2Fcn Plot; hit ANY KEY to Continue')

% "hold" = Retain current graph when adding new graphs

hold on

disp('Hit ANY KEY to show Fill')

pause

%

xn = linspace(xmin, xmax, 500);

fill([xn,fliplr(xn)],[-xn.^2 + 9, fliplr(x.^2 + 1)],[.49 1 .63]), grid on

% alternate RGB triple: [.78 .4 .01]

MATLAB code
Volume Under a Surface
• Use Long Strips to find the Area under a Curve (AuC) by Riemann Summation
• Use Long Boxesto find the VolumeUnder a Surface(VUS) by Riemann Summation
Example  Vol under Surf
• Find the volume under the Surface described by
• Over the Domain
• See Plot at Right
Example  Vol under Surf
• SOLUTION: Find Vol by Double Integral
Example  Vol under Surf
• Completing the Reduction
VUS for NonRectangular Region
• If the Base Region, R, for a Volume Integral is NonRectangular and can be described by InEqualities
• Then by adding up all the long boxes
• If R described by
• Then:
Example  NonRectangular VUS
• Find the volume under the surface
• Over the Region Bounded by
• SOLUTION: First, visualize the limits of integration using a graph of the Base PlaneIntegration Region:
Example  NonRectangular VUS
• The outer limits of integration need to be numerical (no variables), but the Inner limits can contain expressions in x (or y) as in the definition.
• In this case, choose the inner limits to be with respect to y, then find the limits of the y values in terms of x
Example  NonRectangular VUS
• Each y-value in the region is restricted by the constant height 0 at the top, at the bottom by the Line:
• Thus the Double Integral (so far):
• In Simplified Notation
Example  NonRectangular VUS
• Now, Because the outer integral needs to contain only numbers values, consider only the absolute limits on the x-values in the figure:
• a MINimum of 0 and a MAXimum of 5
• Thus the Completed Double Integral
Example  NonRectangular VUS
• Complete the Mathematical Reduction
Example  NonRectangular VUS
• Complete the Mathematical Reduction
• The volume contained underneath the surface and over the triangular region in the XY plane is approximately 69.8 cubic units.
Example  NonRectangular VUS
• Verify Constrained VUS by MuPad

V := int((int(x+E^(x+2*y), y=x-5..0)), x=0..5)

Vnum = float(V)

Average Value
• Recall from Section 5.4 that the average value of a function f of one variable defined on an interval [a, b] is
• Similarly, the average value of a function f of two variables defined on a rectangle R to be:
Example  Average Sales
• Weekly sales of a new product depend on its price p in dollars per item and time t in weeks after its release, can be Modeled by:
• Where S is measured in k-units sold
• Find the average weekly sales of the product during the first six weeks after release and when the product’s price varies between 15 – t and 25 – t.
Example  Average Sales
• SOLUTION: first find the area of the region of integration as shown below
• Note that The price Constraints producea Parallelogram-likeRegion
• By the ParallelogramArea Formula
Example  Average Sales
• Proceed with the Double Integration
Example  Average Sales
• Continue the Double Integration
Example  Average Sales
• Complete the Double Integration
• The average weekly sales is 21,900 units over the time and pricing constraints given.
WhiteBoard Work
• Problems From §7.6
• P7.6-89 → Exposure to Disease
All Done for Today

Volume byRiemannSum

Chabot Mathematics

Appendix

Do On