# Integrating Double Integral of Cos(x+y) from 0 to pi: Step-by-Step Guide

• Math10
In summary, the conversation discusses how to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again, with the given answer being -4. The solution involves using the substitution u=x+y and the trigonometric addition formula sin(a+b) = sin(a)cos(b) + cos(a)sin(b). The final step is to integrate again, similar to the first integral.
Math10

## Homework Statement

How to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again.

## Homework Equations

The answer is -4.

## The Attempt at a Solution

Here's the work:
u=x+y
du=dy
cos(u)du=sin(x+y)
integral of [sin(x+y)] evaluate from 0 to pi dx from 0 to pi
integral of (sin(x+pi)-sin(x))dx from 0 to pi
And what's next?

Math10 said:

## Homework Statement

How to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again.

## Homework Equations

The answer is -4.

## The Attempt at a Solution

Here's the work:
u=x+y
du=dy
cos(u)du=sin(x+y)
integral of [sin(x+y)] evaluate from 0 to pi dx from 0 to pi
integral of (sin(x+pi)-sin(x))dx from 0 to pi
And what's next?

Integrate again, similar to what you did for the first integral.

Er...doesn't the final step that you list give you the answer already?

But I can't. Because sin(x+pi)-sin(x)=sin(x)+sin(pi)-sin(x)=sin(pi)=0. The integral of 0 is?

Math10 said:
But I can't. Because sin(x+pi)-sin(x)=sin(x)+sin(pi)-sin(x)=sin(pi)=0. The integral of 0 is?
##\sin(a+b)\ne \sin a +\sin b##

Math10 said:
Because sin(x+pi)=sin(x)+sin(pi)
That is most certainly not correct. Can you check your compound angle formula?

Then what's sin(a+b)?

@Fightfish , thank you so much for the info! I got it!

## 1. What is integration?

Integration refers to the process of finding the area under a curve on a graph. It is an important concept in calculus and is used to solve various mathematical problems.

## 2. Why is integration important in science?

Integration is important in science because it allows us to determine the total amount or quantity of something, such as the total distance traveled by an object or the total amount of energy produced in a chemical reaction.

## 3. How do you perform integration?

To perform integration, you need to use calculus techniques such as finding the antiderivative of a function or using the fundamental theorem of calculus. This involves breaking down the curve into smaller parts and finding the area under each part, then adding them together to get the total area.

## 4. What are some real-world applications of integration?

Integration has many real-world applications, including calculating the volume of irregularly shaped objects, determining the speed of an object by finding the area under a velocity-time graph, and analyzing data in fields such as physics, engineering, and economics.

## 5. How can I improve my integration skills?

To improve your integration skills, it is important to practice regularly and become familiar with different techniques and applications. You can also seek help from a tutor or join study groups to gain a better understanding of the subject.

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