Q. 6 - Calculus

Question

Discuss the similarities and differences between the definition of the double integral found in Section $13.1$ and the definition of the triple integral found in this section.

Step-by-Step Solution

Answer

Double integrals	Triple integrals
1. It is defined on two coordinate axes usually on $x$ and $y$ axis.	1. It is defined on two coordinate axes  usually on $x,y$ and $z$ axis.
2. The double integral dives the area of a region.	2. The triple integral gives the volume of a solid
3. It is defined as Riemann sum.	3. It is also defined as Riemann sum.
4. The order of evaluation process of double integrals follows from inside to outside.	4. The order of evaluation process of double integrals also follows from inside to outside.

Step 1 . Given information

We need to write the similarities and differences between the definition of the double integral and the definition of the triple integral.

Step 2 . Similarities and differences between the double and the triple integral:

Double integral	Triple integral
1. Let $f\left(x,y\right)$ be a function defined on $R$.  Provided that the limit exists, then the double integral of $f$over $R$ is, $\underset{R}{\iint } f\left(x,y\right)dS=\underset{∆\to 0}{\lim } \sum _{i=1}^l \sum _{j=1}^m f\left(x_i^{*},y_j^{*}\right)∆S$ where the sum in the equation is Riemann sum and $∆=\sqrt{{\left(∆x\right)}^2+{\left(∆y\right)}^2}$.	1. Let $f\left(x,y,z\right)$ be a function defined on $R$. Provided that the limit exists, then the triple integral of $f$ over $R$ is,  $\underset{R}{\iiint } f\left(x,y,z\right)dV=\underset{∆\to 0}{\lim } \sum _{i=1}^l \sum _{j=1}^m \sum _{k=1}^n f\left(x_i^{*},y_j^{*},z_k^{*}\right)∆V$ where the sum in the equation is Riemann sum and $∆=\sqrt{{\left(∆x\right)}^2+{\left(∆y\right)}^2+{\left(∆z\right)}^2}$.
2. It is defined on two coordinates axes usually on $x$ and $y$ axes.	2. It is defined on three coordinate axes usually on $x,y,z$ axes.
3. The double integral gives the area of a region.	3. The triple integral gives the volume of a solid.
4. It is defined as Riemann sum.	4. It is also defined as Riemann sum.
5. The order of evaluation process of double integrals follows from inside to outside.	5. The order of evaluation process of triple integrals also follows from inside to outside.