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3.1: Double Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Multiple_Integrals/3.01%3A_Double_Integrals
Suppose that you want to compute the mass of a plate that fills the region $\mathcal{R}$ in the $xy$ -plane. Suppose further that the density of the plate, say in kilograms per square meter, depend...Suppose that you want to compute the mass of a plate that fills the region $\mathcal{R}$ in the $xy$ -plane. Suppose further that the density of the plate, say in kilograms per square meter, depends on position.
3.4: Green’s Theorem
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_Calculus/3.04%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
2.1: Double Integrals over Rectangular Regions
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_Regions
In this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xyxy-plane. Many of the properties of double integrals are s...In this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xyxy-plane. Many of the properties of double integrals are similar to those we have already discussed for single integrals.
2.1E: Exercises
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_Regions/2.1E%3A_Exercises
This page covers exercises on estimating volumes and integrals using numerical methods like the midpoint rule and Riemann sums for specific functions over defined regions. It includes discussions on d...This page covers exercises on estimating volumes and integrals using numerical methods like the midpoint rule and Riemann sums for specific functions over defined regions. It includes discussions on double integrals, solid geometry, and inequalities related to these integrals. The text also addresses the average value of functions, specifically calculating the average temperature across a rectangular region, with results and estimations presented in data tables.
3.3: Applications of Double Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Multiple_Integrals/3.03%3A_Applications_of_Double_Integrals
Double integrals are useful for more than just computing areas and volumes. Here are a few other applications that lead to double integrals.
2: Multiple Integration
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration
Now we examine integral calculus in multiple dimensions. Just as a partial derivative allows us to differentiate a function with respect to one variable while holding the other variables constant, we ...Now we examine integral calculus in multiple dimensions. Just as a partial derivative allows us to differentiate a function with respect to one variable while holding the other variables constant, we will see that an iterated integral allows us to integrate a function with respect to one variable while holding the other variables constant.
11.5: Double Integrals in Polar Coordinates
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/11%3A_Multiple_Integrals/11.05%3A_Double_Integrals_in_Polar_Coordinates
In particular, the rectangular coordinates of a point $P$ are given by an ordered pair $(x,y)\text{,}$ where $x$ is the (signed) distance the point lies from the $y$ -axis to $P$ and $y$ is...In particular, the rectangular coordinates of a point $P$ are given by an ordered pair $(x,y)\text{,}$ where $x$ is the (signed) distance the point lies from the $y$ -axis to $P$ and $y$ is the (signed) distance the point lies from the $x$ -axis to $P\text{.}$ In polar coordinates, we locate the point by considering the distance the point lies from the origin, $O = (0,0)\text{,}$ and the angle the line segment from the origin to $P$ forms with the positive $x$ -axis.
2.3: Double Integrals in Polar Coordinates
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.03%3A_Double_Integrals_in_Polar_Coordinates
Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept ...Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region.
2.2: Double Integrals over General Regions
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/02%3A_Multiple_Integration/2.02%3A_Double_Integrals_over_General_Regions
In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need t...In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.
3.1: Double Integrals
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.01%3A_Double_Integrals
In single-variable calculus, differentiation and integration are thought of as inverse operations. There is a similar way of defining integration of real-valued functions of two or more variables? Rec...In single-variable calculus, differentiation and integration are thought of as inverse operations. There is a similar way of defining integration of real-valued functions of two or more variables? Recall also that the definite integral of a nonnegative function f(x)≥0 represented the area “under” the curve y=f(x). As we will now see, the double integral of a nonnegative real-valued function f(x,y)≥0 represents the volume “under” the surface z=f(x,y).

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