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- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.05%3A_Cylindrical_and_Spherical_CoordinatesIn this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are u...In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.07%3A_Calculating_Centers_of_Mass_and_Moments_of_InertiaIn this section we develop computational techniques for finding the center of mass and moments of inertia of several types of physical objects, using double integrals for a lamina (flat plate) and tri...In this section we develop computational techniques for finding the center of mass and moments of inertia of several types of physical objects, using double integrals for a lamina (flat plate) and triple integrals for a three-dimensional object with variable density. The density is usually considered to be a constant number when the lamina or the object is homogeneous; that is, the object has uniform density.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.08%3A_Change_of_Variables_in_Multiple_IntegralsWhen solving integration problems, we make appropriate substitutions to obtain an integral that becomes much simpler than the original integral. We also used this idea when we transformed double inte...When solving integration problems, we make appropriate substitutions to obtain an integral that becomes much simpler than the original integral. We also used this idea when we transformed double integrals in rectangular coordinates to polar coordinates and transformed triple integrals in rectangular coordinates to cylindrical or spherical coordinates to make the computations simpler.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.09%3A_Chapter_2_Review_ExercisesThis page presents exercises in multivariable calculus covering integral evaluation, area and volume determination, and center of mass calculations. Problems include confirmations of Fubini's theorem,...This page presents exercises in multivariable calculus covering integral evaluation, area and volume determination, and center of mass calculations. Problems include confirmations of Fubini's theorem, double and triple integrals, and practical applications like Earth modeling and ski resort work estimation. Key answers for selected tasks illustrate results associated with integrals and geometric figures.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.04%3A_Triple_Integrals/2.4E%3A_ExercisesIf the charge density at an arbitrary point \((x,y,z)\) of a solid \(E\) is given by the function \(\rho (x,y,z)\), then the total charge inside the solid is defined as the triple integral \(\displays...If the charge density at an arbitrary point \((x,y,z)\) of a solid \(E\) is given by the function \(\rho (x,y,z)\), then the total charge inside the solid is defined as the triple integral \(\displaystyle \iiint_E \rho (x,y,z) \,dV.\) Assume that the charge density of the solid \(E\) enclosed by the paraboloids \(x = 5 - y^2 - z^2\) and \(x = y^2 + z^2 - 5\) is equal to the distance from an arbitrary point of \(E\) to the origin.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.06%3A_Triple_Integrals_in_Cylindrical_and_Spherical_Coordinates/2.6E%3A_ExercisesThis page presents exercises focused on evaluating triple integrals over solid regions in three-dimensional space, using cylindrical and spherical coordinates. It includes function transformations, co...This page presents exercises focused on evaluating triple integrals over solid regions in three-dimensional space, using cylindrical and spherical coordinates. It includes function transformations, computations of volumes for shapes like cylinders and cones, and integral evaluations, providing explicit examples and solutions. Additionally, the document emphasizes the conversion between coordinate systems and discusses the properties of continuous functions with symmetry.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.03%3A_Double_Integrals_in_Polar_Coordinates/2.3E%3A_ExercisesThis page offers a collection of exercises and solutions focused on evaluating double integrals using polar coordinates. It covers conversions from rectangular to polar coordinates, area and volume ca...This page offers a collection of exercises and solutions focused on evaluating double integrals using polar coordinates. It covers conversions from rectangular to polar coordinates, area and volume calculations under various geometric shapes (like cones and spheres), and the properties of radial functions. Key topics include evaluating integrals, calculating areas and volumes, and understanding joint density functions associated with normal distributions.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.07%3A_Calculating_Centers_of_Mass_and_Moments_of_Inertia/2.7E%3A_ExercisesThis page presents exercises focused on calculating mass, center of mass, and moments of inertia for shapes such as triangles, rectangles, and three-dimensional solids like cylinders and hemispheres u...This page presents exercises focused on calculating mass, center of mass, and moments of inertia for shapes such as triangles, rectangles, and three-dimensional solids like cylinders and hemispheres using specified density functions. It includes structured problems with calculated answers and emphasizes the use of triple integrals, moments about axes, and radii of gyration.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.02%3A_Double_Integrals_over_General_Regions/2.2E%3A_ExercisesThis page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical...This page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical exercises on double integrals and explores geometric interpretations in three-dimensional space. Additionally, it discusses the lunes of Alhazen, proving their area is equivalent to that of a corresponding triangle.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_Regions/2.1E%3A_ExercisesThis 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.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_RegionsIn 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.
