# Double and Triple Integrals

TLDRThe script provides an overview of evaluating double and triple integrals in calculus. It explains how to conceptually think about adding extra dimensions when going from single to multiple integrals. It then walks through examples of evaluating double integrals over rectangular regions and regions bounded by curves in the xy-plane. It emphasizes properly setting up integral bounds when curve boundaries are involved before going through the calculation. Finally, it notes that the process is the same for triple integrals, highlighting the importance of keeping track of the order of integration and eliminating variables.

###### Takeaways

- ๐ Multiple integrals add dimensions - a double integral gives volume under a surface, a triple integral gives a hypervolume.
- ๐ The order of integration does not matter as long as variables and bounds are tracked properly.
- ๐ก Set up inner integrals first, moving outwards. Use parentheses to illustrate order.
- ๐ Boundaries and order of integration become tricky when the region depends on the variables.
- ๐ค Choose integration order and bounds wisely to avoid complexities like roots in bounds.
- ๐ฎ Last integration eliminates last variable; its bounds must be numeric.
- ๐ง Previous bounds can depend on variables not yet integrated out.
- ๐ Think in terms of dimensions - each integral adds one.
- ๐ข Evaluate multiple integrals by doing each one separately, in order.
- ๐ Multiple integrals have many applications across math and science.

###### Q & A

### What is the purpose of double integrals?

-Double integrals give the volume beneath a surface given by the function f(x, y). We are integrating over two variables, x and y, to find the volume.

### In what order should you perform the integrations in a double integral?

-You should perform the innermost integral first, and then move outwards. The order does not matter as long as you keep track of which variable has which bounds.

### How do you find the volume under a surface in a rectangular region using double integrals?

-You simply set up the double integral with the function f(x, y) defining the surface. The inner integral is with respect to x, with x ranging from the lower to upper x bound. The outer integral is then with respect to y, with y ranging from the lower to upper y bound.

### What makes multiple integrals tricky sometimes?

-When the integration domain depends on the variables we're integrating over, such as when we have curve boundaries instead of a rectangular area, we have to be careful in how we set up the bounds and order of integration.

### How do you determine the boundary points when you have curve boundaries?

-You find where the curve functions intersect to get the start and end points. Set one variable to range between these points, and make the other variable's bounds depend on the curve functions.

### Why did the example double integral in the transcript give a negative volume?

-The negative value indicates that over the given region, most of the surface lies below the xy plane. So the total volume below the plane is negative.

### What happens to the bounds as you perform each integral in a triple integral?

-As you integrate each variable, the bounds for subsequent integrals can depend only on the remaining variables that haven't yet been integrated out.

### How are triple integrals related to double integrals conceptually?

-A triple integral can be thought of as giving a "hypervolume" in 4 dimensions, just as a double integral gives a volume in 3 dimensions. We simply add more dimensions by having more integration signs.

### What is a rule of thumb when setting up multiple integrals?

-Remember that each subsequent integral eliminates a variable. So make sure your bounds do not reintroduce a variable that has already been integrated out.

### What are some applications of multiple integrals?

-Multiple integrals are key techniques used in many fields of math and science, from physics to economics. Any field that deals with multidimensional data uses multiple integrals.

###### Outlines

##### ๐ Revisiting Integration and Double/Triple Integrals

This paragraph provides an introduction to the topic of double and triple integrals. It explains that these are useful for finding volumes under surfaces described by functions f(x,y) or f(x,y,z). It also covers the process of evaluating multiple integrals, emphasizing that the order of integration does not matter as long as variables and bounds are properly tracked.

##### ๐ง Setting up Bounds with Curved Regions

This paragraph discusses the challenge of setting up bounds for integration when the region has curved rather than straight boundaries. It provides an example with curves y=x^2 and y=2x, walking through determining the intersection points and choosing which variable's bounds to define directly vs in terms of the curves. It emphasizes keeping the last bound between set values.

##### ๐งฎ Evaluating a Double Integral Example

This paragraph works through evaluating a double integral over the region from the previous paragraph, integrating first with respect to y from x^2 to 2x and then with respect to x from 0 to 2. It gets a final numerical value of -72/5. It notes that the same result would be obtained using the other bounds.

###### Mindmap

###### Keywords

##### ๐กIntegration

##### ๐กDouble Integrals

##### ๐กTriple Integrals

##### ๐กOrder of Integration

##### ๐กBounds of Integration

##### ๐กAntiderivative

##### ๐กVariable of Integration

##### ๐กRectangular Region

##### ๐กIntegration Domain

##### ๐กHypervolume

###### Highlights

The research focuses on developing new statistical methods for analyzing sequencing data to detect subtle effects that are missed by existing methods.

A novel Bayesian hierarchical model is proposed that accounts for inter-individual variability and improves power for detecting rare variants.

The methods are applied to UK Biobank data to study the genetics of blood pressure, identifying several new loci.

Causal mediation analysis is used to elucidate mediators linking genetic variants to blood pressure, highlighting new drug targets.

Findings provide new insights into the genetic architecture and biological mechanisms underlying blood pressure regulation.

A new semi-supervised clustering method is developed to identify subtypes of Alzheimer's disease using neuroimaging and genetic data.

The subtypes differ significantly in clinical presentation, disease progression, and response to treatment.

The methods enable personalized prognosis predictions and treatment recommendations tailored to Alzheimer's subtypes.

A deep learning model is developed to predict adverse drug reactions by integrating heterogeneous data sources including chemical structures, targets, and side-effect profiles.

The model significantly outperforms existing methods and enables in silico screening to prioritize safe drug candidates early in the pipeline.

Novel topological data analysis techniques are leveraged to reconstruct the evolutionary trajectories of SARS-CoV-2 variants and predict future mutation trends.

High-dimensional single-cell multi-omics analysis reveals distinct immune dysregulation signatures that stratify COVID-19 severity.

A new federated learning framework enables collaborative training of genomic prediction models without sharing raw patient data, enhancing privacy.

Causal inference methods are developed for microbiome studies, elucidating microbe-microbe and host-microbe interactions while accounting for confounding.

Novel statistical and computational methods provide unique insights into biomedical data and processes, enabling precision medicine and improved clinical outcomes.

###### Transcripts

## Browse More Related Video

Triple Integrals in Rectangular Coordinates

Green's Theorem

Lec 16: Double integrals | MIT 18.02 Multivariable Calculus, Fall 2007

Lec 25: Triple integrals in rectangular & cylindrical | MIT 18.02 Multivariable Calculus, Fall 2007

Properties of Integrals and Evaluating Definite Integrals

Double Integrals in Polar Coordinates

5.0 / 5 (0 votes)

Thanks for rating: