Calculus Double Integrals Overview

The integral where many variables are involved motivates the study of integral calculus of several variables.

A function f(x, y) that has a single value and is continuous within a closed curve region.

A sum where the area of each sub-region is multiplied by the function's value at a point in that sub-region.

The limit of the sum as the number of sub-regions approaches infinity and each sub-region's area approaches zero.

It represents the double integral of f(x, y) over the region R.

Integrate f(x, y) first with respect to x (treating y as a constant), then with respect to y.

The limits of y vary from f1(x) to f2(x), while x varies from a to b. This defines the region R.

The region of integration is divided into vertical strips, representing the order of integration (dy dx).

The order of integration in a double integral determines the order in which we integrate with respect to each variable. Changing the order of integration can sometimes make the evaluation of the integral easier.

When changing the order of integration, we need to identify the limits of integration for the new order. This involves finding the equations of the bounding curves of the region of integration.

Draw a horizontal strip on the region to indicate the direction of integration for 𝑑𝑦𝑑𝑥. The horizontal strip should be parallel to the y-axis.

The limits for the inner integral are the equations of the curves that intersect the horizontal strip. These curves will be functions of the outer variable.

The limits for the outer integral are the constants that define the range of the outer variable. These constants are determined by the endpoints of the region along the outer variable's axis.

When evaluating a double integral, the process involves integrating with respect to the inner variable first, followed by integrating with respect to the outer variable.

The integration of a double integral is used to find the area of the region that is the result of the intersection of the functions.

To divide the region into multiple parts for integrating, create multiple subregions and calculate their areas by using the same method as before. Then, add the areas of each subregion to find the total area of the region.

The process of switching the order in which you integrate a multivariable function, where the inner integral is evaluated first, followed by the outer integral.

The area of a region defined by a polar curve is found by integrating over the region. This involves integrating over the angles and then integrating over the radius for each angle.

A graphical representation of a function's domain of integration, showing the boundaries of the area being integrated over.

A line or curve that defines one of the boundaries of the region of integration.

The limits of integration for the angle (θ) are determined by the range of the polar curve. This range is the interval over which the curve is defined.

A horizontal slice across the region of integration, used to visualize the order of integration when the outer integral is with respect to x.

The limits of integration for the radius (r) are defined by the curve itself at each angle. They represent the boundaries between the inside and outside edges of the region.

A vertical slice across the region of integration, used to visualize the order of integration when the outer integral is with respect to y.

The integrand in polar coordinates is r, representing the distance from the origin. This is multiplied by the infinitesimally small area element rdrdθ.

A technique used to calculate the integral of a multivariable function by first integrating with respect to one variable and then with respect to the other.

The area of a cardioid shaped polar curve is calculated by integrating the equation that describes the cardioid over the specified range of angles.

The area of a circle in polar coordinates is calculated by integrating the equation of the circle over the range of angles that define the full circle.

The process of switching the order of integration in a double integral, where the original order is dy dx and the changed order is dx dy.

The new limits of integration for x when changing the order of integration from dy dx to dx dy.

A lemniscate is a figure-eight shaped polar curve. Its area is found by integrating the equation of the curve over the appropriate angle range.

Double integration is the process of integrating a function over a two-dimensional region. In polar coordinates, this involves integrating over both angle and radius.

The new limits of integration for y when changing the order of integration from dy dx to dx dy.

The calculation of a double integral after changing the order of integration, allowing for easier evaluation of the integral.

The process of evaluating the inner integral first, treating the other variable as a constant, before evaluating the outer integral.

An expression used to represent the area of the region of integration, often calculated through integration.

A double integral where the order of integration is changed from dy dx to dx dy, altering the way the region of integration is sliced.

The process of carefully selecting and modifying the limits of integration for each variable after changing the order of integration.

An integral where the function being integrated depends on two or more variables, requiring integration with respect to each variable.

The integral ∫r^2*f(r,θ)dr represents integration along a radial line (PQ), where r varies and θ remains constant. This integration measures the area of a strip along the radial direction.

The integral ∫𝜃1^𝜃2*f(r, θ)dθ represents integration along the arc of a circle (AC to BD), where θ varies and r remains constant. This integral measures the area of a strip along the angular direction.

The double integral in polar coordinates ∫𝜃1^𝜃2 ∫r1^r2 f(r, θ) dr dθ represents the integration of a function f(r,θ) over a region in polar coordinates. This region is defined by the limits of integration for r and θ, which represents the change in radius and angle, respectively.

Changing the order of integration in a double polar integral changes the order in which you integrate. For the double integral, ∫𝜃1^𝜃2 ∫r1^r2 f(r, θ) dr dθ, the order of integration is first along the radius (dr) and then along the arc (dθ).

When you change the order of integration in a double integral, you need to adjust the limits of integration to reflect the new order. For example, in ∫𝜃1^𝜃2 ∫r1^r2 f(r, θ) dr dθ you have to readjust the limits of r and θ to ensure that the correct area is covered.

The integral ∫𝜃1^𝜃2 ∫r1^r2 r dr dθ is used to calculate the area of a region in polar coordinates. The factor 'r' represents the Jacobian determinant which accounts for the transformation from Cartesian to polar coordinates.

The integral often involves the evaluation of trigonometric functions as the limits of integration are typically expressed in terms of angles. The function f(r, θ) is often a combination of trigonometric functions and powers of r.

Sometimes the given limits of integration for θ may be from 0 to π/2. This means that the area being integrated covers an angle range of 90 degrees.

Evaluating integrals in polar coordinates can involve integrating expressions containing cos^n(θ) or sin^n(θ) and solving trigonometric integrals using standard techniques and trigonometric identities.

In some examples, the limits of integration for θ might be from 0 to 2π. This means the area being integrated covers a full rotation of 360 degrees.

Sometimes the limits of integration for θ might be from - π/2 to π/2. This indicates integrating over an area spanning 180 degrees.

To find the area of a region in polar coordinates, you need to calculate the double integral, and the limits of integration often involve trigonometric functions.

The integral ∫𝑎(1−𝑐𝑜𝑠𝜃) 𝑟 2 𝑑𝜃𝑑𝑟 represents integrating over a region defined by the curve r = a(1 - cos θ) and the interval 0 < θ < π/2.

The limits of integration for both r and θ need to be carefully chosen to ensure that the correct area is covered by the integration process.

The integral ∫−𝜋/2^𝜋/2 ∫0^2𝑐𝑜𝑠𝜃 𝑟 2 𝑑𝜃𝑑𝑟 represents integrating over a region bounded by the curve 𝑟 = 2𝑐𝑜𝑠𝜃 and the interval -π/2 < θ < π/2. This effectively integrates over a semicircle.

The integral of a function f(x, y, z) over a three-dimensional region R is defined as ∭R f(x, y, z)dxdydz, where the integration is performed over the volume of the region R.

A type of triple integral where the limits of integration are defined by constant values for each variable, allowing for systematic evaluation.

A three-dimensional region defined by a set of inequalities that restrict the values of each variable. This helps define the boundaries over which the integral is evaluated.

The order of integration in a triple integral determines the sequence in which the variables are integrated. This affects how you write the limits of integration for each variable.

The process of evaluating a triple integral involves integrating the function with respect to each variable one at a time, while treating the other variables as constants. The final result is a single number.

The evaluation of a triple integral where the integrand is a simple function that can be integrated directly. This process involves applying the fundamental theorem of calculus repeatedly.

The concept of iterated integrals allows us to evaluate triple integrals by performing a series of single-variable integrals within a defined order. This simplifies the integration process.

A technique used to simplify the evaluation of triple integrals by transforming the original integral into a new integral with simpler limits of integration. This often involves changing the coordinate system.

The process of converting a triple integral from Cartesian coordinates to a different coordinate system, such as cylindrical or spherical coordinates. This often simplifies the integral's form and makes evaluation easier.

The integral of a function over a volume in space, where the region of integration is defined by a set of inequalities and the integrand is a function of three variables.

The use of triple integrals to solve problems in physics and engineering, such as finding volumes, masses, moments, and centers of gravity.

When a triple integral is used to calculate the volume of a three-dimensional region, the integrand is set to 1, resulting in the integral representing the volume enclosed by the boundaries of the region.

The use of triple integrals to find the mass of a three-dimensional object. The integrand is the density function, representing the mass per unit volume.

The use of triple integrals to calculate the center of mass of a three-dimensional object. The integrand is the density function, and the coordinates of the center of mass are calculated as weighted averages.

The use of triple integrals to calculate moments of inertia of a three-dimensional object. The integrand is the density function multiplied by the square of the distance from a given axis.