3A: Double Integrals

In part A, we will learn about double integration over regions in the plane. Conceptually an integral is a sum. We will apply this idea to computing the mass, center of mass and moment of inertia of a two dimensional body and the volume of a region bounded by surfaces.

In order to compute double integrals we will have to describe regions in the plane in terms of the equations describing their boundary curves. After that, the computation just becomes two single variable integrations done iteratively.

Last Modification: 2020-09-17 Thu 12:23

Captured On: [2020-01-18 Sat]
Source: Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

Session 47: Definition of Double Integration

Captured On [2020-02-06 Thu 13:14] Source Session 47: Definition of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall integrals of 1 variable function Figure 2: Meaning of double integral Figure 3: Small pieces of area Figure 4: Computing double integrals taking slices Figure 5: Iterated integrals 2 Can we compute a double integrals as a limit of Riemann sums?...

Session 48: Examples of Double Integration

Captured On [2020-02-06 Thu 13:14] Source Session 48: Examples of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Example of iterated integral Figure 2: Inner integral Figure 3: Outer integral Figure 4: Same function, other region Figure 5: Inner function, more complex region Figure 6: Inner cont., outer function...

Session 49: Exchanging the Order of Integration

Captured On [2020-02-06 Thu 13:14] Source Session 49: Exchanging the Order of Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Exchanging order of integration Figure 2: Example of exchanging order of integration Figure 3: Example conclusion 2 How can we exchanging the order of integration? 2.1 Front How can we exchanging the order of integration?...

Session 50: Double Integrals in Polar Coordinates

Captured On [2020-02-06 Thu 13:15] Source Session 50: Double Integrals in Polar Coordinates | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Double integral as interated integral $\dd{y}\dd{x}$ Figure 2: Function in polar coordinates Figure 3: Setting up interated integrals on polar coordinates (1) Figure 4: Relationship between $dA$ and $d\theta$ and $dr$...

Session 51: Applications: Mass and Average Value

Captured On [2020-02-06 Thu 13:15] Source Session 51: Applications: Mass and Average Value | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area of a region $R$ Figure 2: Mass of a flat object with density $\delta$ Figure 3: Average value of $f$ Figure 4: Center of mass of a (planar) object with density $\delta$...

Session 52: Applications: Moment of Inertia

Captured On [2020-02-06 Thu 13:15] Source Session 52: Applications: Moment of Inertia | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Moment of inertia Figure 2: Kinetic energy Figure 3: Moment of inertia for each $\Delta A$ Figure 4: Moment of inertia about the origin Figure 5: Moment of inertia about $x\text{-axis}$...

Session 53: Change of Variables

Captured On [2020-02-06 Thu 13:16] Source Session 53: Change of Variables | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Change variable in double integral Figure 2: Basic example Figure 3: Basic example continuation Figure 4: Find scaling factor Figure 5: Linear change of variables Figure 6: Example of linear change of variable...

Session 54: Example: Polar Coordinates

Captured On [2020-02-06 Thu 13:16] Source Session 54: Example: Polar Coordinates | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Polar coordinates Figure 2: Scaling factor Figure 3: Recall reciprocal rule for partial derivatives 2 Proof $dA$ in polar coordinates with the Jacobian 2.1 Front Proof $\dd{A}$ in polar coordinates with the Jacobian...

Session 55: Example

Captured On [2020-02-06 Thu 13:16] Source Session 55: Example | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area element Figure 2: Integrand in terms of $u,v$ Figure 3: Bounds Figure 4: Bounds switching to $uv$ picture 2 How can we compute $\Delta A$ in another $u,v$ coordinates 2.1 Front How can we compute $\Delta A$ in another $u,v$ coordinates...