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Chabot Mathematics. §7.6 Double Integrals. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer 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

Licensed Electrical & Mechanical



Review §
  • Any QUESTIONS About
    • §7.5 → Lagrange Multipliers
  • Any QUESTIONS About HomeWork
    • §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 integ1
  • 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 integ2
  • Performing The AntiDerivation while not including the Constant find:
  • Now Let:
  • Then substitute, then multiply by ∂x
partial deriv partial integ3
  • Integrating find:
  • After AntiDerivation:
  • But ReCall:
  • Back Substituting find
  • By the Associative Property
partial deriv partial integ4
  • 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
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 curves1
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 curves2
Area BETWEEN Curves
  • Substitute:
  • Then
  • Next Add Up all the Strips to find the Total Area, A
area between curves3
Area BETWEEN Curves
  • This Relation
  • Is simply a Riemann Sum
  • Then in the Limit
  • Find
example area between curves
Example  Area Between Curves
  • Find the area of the region contained between the parabolas
example area between curves1
Example  Area Between Curves
  • SOLUTION: Use Double Integration
matlab code

% 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')



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]

volume under a surface
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
Example  Vol under Surf
  • Find the volume under the Surface described by
  • Over the Domain
  • See Plot at Right
example vol under surf1
Example  Vol under Surf
  • SOLUTION: Find Vol by Double Integral
example vol under surf2
Example  Vol under Surf
  • Completing the Reduction
vus for nonrectangular region
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
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 vus1
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 vus2
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 vus3
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 vus4
Example  NonRectangular VUS
  • Complete the Mathematical Reduction
example nonrectangular vus5
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 vus6
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
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
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 sales1
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 sales2
Example  Average Sales
  • Proceed with the Double Integration
example average sales3
Example  Average Sales
  • Continue the Double Integration
example average sales4
Example  Average Sales
  • Complete the Double Integration
  • The average weekly sales is 21,900 units over the time and pricing constraints given.
whiteboard work
WhiteBoard Work
  • Problems From §7.6
    • P7.6-89 → Exposure to Disease
all done for today
All Done for Today

Volume byRiemannSum


Chabot Mathematics


Do On


Bruce Mayer, PE

Licensed Electrical & Mechanical

7 3 learning goals
§7.3 Learning Goals
  • 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