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# Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] - PowerPoint PPT Presentation

Chabot Mathematics. §7.6 Double Integrals. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] 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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§7.6 DoubleIntegrals

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

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

• 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

• 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

• Substitute:

• Then

• Next Add Up all the Strips to find the Total Area, A

• 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

% 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

• 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)

• 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

Volume byRiemannSum

Appendix

Do On

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

• Deﬁne and compute double integrals over rectangular and NONrectangular regions in the xy plane

• Use double integrals in problems involving

• Area

• Volume,

• Average Value

• Population Density