# Evaluate the following double integral

• brendan_foo
In summary, The conversation discusses a double integral, with the first integral being computed successfully using parts. However, integrating with respect to the second variable, x, poses difficulties. Different techniques are suggested, such as interchanging the order of integration and integrating with respect to y first, or using rotation of variables. Ultimately, the conversation ends with the individual expressing frustration and requesting hints or tips for solving the integral.
brendan_foo
Just had an exam and I had to evaluate the following double integral, with limited success

$$\int_0^1 \int_0^{\pi} y\sin(xy) {dy} {dx}$$

I managed to compute the first integral, that was ok, using parts. But trying to integrate that with respect to dx just yielded a whole lot of trouble. Could someone have a skim over this and see if its do-able using elementary calculus procedures {I say elementary, but you know what I mean}.

Thankyou

I would interchange the order of integration and integrate y Sin(x y) with respect to x to get -Cos(x y). Should be easy from here.

Rotation of variables was not covered at all... So say you weren't armed with that tool, what then? (not a cop out, honestly)

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Part integration wrt "y" and then integrate the result wrt "x",what else...?

Daniel.

The second integration w.r.t x wouldn't work for me... Right I am going to write down what I've done.

For the first inner integration, i have as follows:

$$\int_0^{\pi} y\cdot \sin(xy) dy = -\frac{\pi}{x}\cos({\pi x}) + \frac{1}{x^2} \cdot \sin({\pi x})$$

I have tried with an abundance of attempts to evaluate all that as an integral with respect to x and I go nowhere conclusive, and its really starting to irritate me. Man, i suck at this.

Any hints or tips...please!

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## 1. What is a double integral?

A double integral is a type of mathematical operation that involves finding the volume under a two-dimensional surface. It is similar to a regular integral, but instead of finding the area under a curve, it finds the volume under a surface.

## 2. How is a double integral evaluated?

A double integral is evaluated by breaking it down into smaller, simpler integrals. This can be done by using the concepts of Fubini's theorem and the properties of integrals such as linearity and substitution. The limits of integration can also be changed to simplify the integral.

## 3. What are the uses of double integrals?

Double integrals have various uses in physics, engineering, and other fields of science. They are used to find the area, volume, and mass of objects with irregular shapes, calculate moments of inertia, and solve problems involving partial derivatives.

## 4. What are the different types of double integrals?

There are two types of double integrals: iterated integrals and double integrals with polar coordinates. In iterated integrals, the limits of integration are constants, while in double integrals with polar coordinates, the limits are given by polar equations.

## 5. What are some common mistakes when evaluating double integrals?

One common mistake is mixing up the order of integration, which can lead to incorrect answers. Another mistake is forgetting to include the appropriate measure, such as dx or dy, when converting to polar coordinates. It is also important to carefully choose the limits of integration to avoid over or underestimating the integral.

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