9.3 Fubini's theorem and iterated integrals

Fubini's theorem and iterated integrals are game-changers for double integrals. They let us break down complex two-dimensional problems into simpler one-dimensional calculations, making our lives way easier.

This powerful tool connects to the broader concept of double integrals over rectangular regions. It's like having a secret weapon that simplifies tricky integrals, letting us tackle real-world problems in physics and engineering with more confidence.

Fubini's Theorem and Iterated Integrals

Definition and Concept of Fubini's Theorem

Top images from around the web for Definition and Concept of Fubini's Theorem

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Concept of Fubini's Theorem

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

1 of 3

• States that if $f(x,y)$ is continuous over a closed, bounded region $R$, then the double integral of $f$ over $R$ equals the iterated integral of $f$ over $R$
• Allows for the computation of a double integral by iterating the integration process, integrating with respect to one variable at a time
• Provides a way to evaluate double integrals by reducing them to repeated single integrals
• Useful for simplifying complex double integrals into more manageable single integrals

Iterated Integrals and Interchange of Integration Order

• An iterated integral is a repeated integral, integrating with respect to one variable at a time
  • For example, $\int_a^b \int_c^d f(x,y) dy dx$ is an iterated integral, integrating first with respect to $y$ and then with respect to $x$
• Fubini's theorem allows for the interchange of integration order under certain conditions
  • If $f(x,y)$ is continuous over the region $R$, then $\int_R f(x,y) dA = \int_a^b \int_c^d f(x,y) dy dx = \int_c^d \int_a^b f(x,y) dx dy$
• The order of integration can be changed without affecting the value of the double integral
  • This is useful when one order of integration is easier to evaluate than the other

Representing Double Integrals as Repeated Single Integrals

• Fubini's theorem allows for the representation of a double integral as repeated single integrals
• The double integral $\iint_R f(x,y) dA$ can be written as an iterated integral $\int_a^b \int_c^d f(x,y) dy dx$ or $\int_c^d \int_a^b f(x,y) dx dy$
  • The inner integral is evaluated first, treating the other variable as a constant
  • The result of the inner integral is then integrated with respect to the outer variable
• This representation simplifies the evaluation of double integrals by breaking them down into single integrals
  • For example, $\iint_R xy dA$ over the region $R = \{(x,y) | 0 \leq x \leq 1, 0 \leq y \leq x\}$ can be evaluated as $\int_0^1 \int_0^x xy dy dx$

Conditions for Fubini's Theorem

Continuity Requirement

• Fubini's theorem requires the function $f(x,y)$ to be continuous over the region of integration
• Continuity ensures that the function has no gaps or breaks within the region
  • A function is continuous if it has no jumps or holes in its graph
• If the function is not continuous, Fubini's theorem may not hold, and the iterated integrals may not equal the double integral
  • For example, if $f(x,y) = \frac{xy}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$, then $f$ is not continuous at $(0,0)$, and Fubini's theorem does not apply

Bounded Region Requirement

• Fubini's theorem also requires the region of integration to be bounded
• A bounded region is a closed and finite region in the $xy$-plane
  • It can be described by a set of inequalities, such as $a \leq x \leq b$ and $c \leq y \leq d$
• If the region is unbounded, Fubini's theorem may not hold, and the iterated integrals may not converge
  • For example, the region $R = \{(x,y) | x > 0, y > 0\}$ is unbounded, and Fubini's theorem cannot be applied to integrals over this region

Applications

Applications in Physics

• Double integrals and Fubini's theorem have various applications in physics
• Calculating moments of inertia of two-dimensional objects
  • The moment of inertia measures an object's resistance to rotational acceleration and depends on the object's mass distribution
  • Double integrals are used to integrate the product of the mass density and the square of the distance from the axis of rotation over the object's area
• Determining the center of mass of two-dimensional objects
  • The center of mass is the point where the object's total mass can be considered concentrated
  • Double integrals are used to integrate the product of the mass density and the position coordinates over the object's area, divided by the total mass

Applications in Engineering

• Double integrals and Fubini's theorem are also useful in engineering applications
• Calculating the volume of solid objects
  • Double integrals can be used to integrate the cross-sectional area of an object along its length to determine its volume
  • For example, the volume of a cylinder can be calculated by integrating the area of a circle along the cylinder's height
• Determining the average value of a function over a two-dimensional region
  • Double integrals are used to integrate the function over the region and divide by the area of the region
  • This is useful in heat transfer problems, where the average temperature over a surface is of interest