Solving a Double Integral: 0.23 Numerically, Analytically Possible?

[LagrangeEuler]

Homework Statement

Solve double integral
\int^1_0\int^1_x\sin(y^2)dydx

Homework Equations

The Attempt at a Solution

I got with Wolfram Mathematica 7.0 result 0.23 numerically. Can it be solved analyticaly?

 

[SammyS]

Homework Statement

Solve double integral
\int^1_0\int^1_x\sin(y^2)dydx

Homework Equations

The Attempt at a Solution

I got with Wolfram Mathematica 7.0 result 0.23 numerically. Can it be solved analyticaly?

Yes, it can be solved analytically. Change the order of integration.

 

[LagrangeEuler]

I'm not sure how?

 

[Michael Redei]

At present, you're integrating over a set of points (x,y) with

(1) 0 ≤ x ≤ 1
(2) x ≤ y ≤ 1

If you're going to reverse the order of integration, you need two new restraints:

(1') (some number) ≤ y ≤ (some other number)
(2') (some number or an expression with y) ≤ x ≤ (another expression that may contain y)

Try sketching this set on a piece of paper and translate (1),(2) to a geometric shape and that back again to (1'),(2'). Then you can rewrite your integral:
$$
\int_{(1)}\int_{(2)} \sin(y^2)\, \mathrm dy \, \mathrm dx = \int_{(1')}\int_{(2')}\sin(y^2) \, \mathrm dx \, \mathrm dy
$$

 

[Dick]

I'm not sure how?

Draw the region you are integrating over. It's a triangle in the xy plane, right? Then just set the integration up so you do dx first then dy.

 

[Mark44]

I'm not sure how?

Draw the region you are integrating over. It's a triangle in the xy plane, right? Then just set the integration up so you do dx first then dy.

To add to what Dick said, whenever a situation arises where you're considering changing the order of integration, it's alway a good idea to sketch the region over which integration is taking place.