Changing the Order of Integration for Double Integrals

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In summary, the given integral can be rewritten as the integral of y*sin(y^2) from 0 to sqrt(pi), by reversing the order of the integrals. This simplifies the computation by allowing for a simple substitution.
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that is [tex]\int_0^{\sqrt{\pi}}\int_x^{\sqrt{\pi}} \sin(y^2) ~dy ~dx[/tex]

Reverse the order of the integrals (which is possible since the integrand is positive) :
[tex]0\leq x\leq y\leq \sqrt{\pi} \Rightarrow [/tex] y ranges from 0 to [tex]\sqrt{\pi}[/tex]

[tex]0\leq x\leq y\leq \sqrt{\pi} \Rightarrow [/tex] x varies from 0 to y.

So the integral is now :

[tex]\int_0^{\sqrt{\pi}}\left(\int_0^y \sin(y^2) ~dx\right)~dy[/tex]

[tex]=\int_0^{\sqrt{\pi}}\left(\sin(y^2)\int_0^y dx\right)~dy[/tex]
 
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Because you changed the order of integration, you don't have to compute the integral of sin(y^2) but rather y*sin(y^2), which can be done by a simple substitution.
 

What is a "compute integral"?

A compute integral is a mathematical concept used in calculus to calculate the area under a curve. It is represented by the symbol ∫ and is used to find the total change or accumulation of a quantity over a given interval.

What is the process for computing an integral?

The process for computing an integral involves finding the antiderivative of a function and evaluating it at the upper and lower limits of the interval. This is known as the Fundamental Theorem of Calculus and allows us to find the total area under a curve.

Why is computing integrals important?

Computing integrals is important because it allows us to solve a variety of real-world problems, such as calculating the distance traveled by an object or the total volume of a three-dimensional shape. It is also an essential tool in physics, engineering, and economics.

What are the different types of integrals?

There are two main types of integrals: definite and indefinite. A definite integral has specific upper and lower limits and gives a specific numerical value for the area under the curve. An indefinite integral does not have limits and gives a general solution in terms of a constant.

What are some techniques for computing integrals?

Some techniques for computing integrals include substitution, integration by parts, and trigonometric substitution. These techniques allow us to simplify complicated integrals and make them easier to solve.

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