Simple Nature - Light and Matter

differential. The result isarea ===== a∫ b∫ ay=0 x=0∫ b ∫ ay=0∫ by=0∫ by=0∫ by=0x=0(∫ aa dy= ab .dydAdx dydxx=0)dyArea of a triangle example 18⊲ Find the area of a 45-45-90 right triangle having legs a.⊲ Let the triangle’s hypotenuse run from the origin to the point(a, a), and let its legs run from the origin to (0, a), and then to(a, a). In other words, the triangle sits on top of its hypotenuse.Then the integral can be set up the same way as the one before,but for a particular value of y, values of x only run from 0 (on they axis) to y (on the hypotenuse). We then havearea ====∫ a∫ yy=0 x=0∫ a ∫ yy=0∫ ay=0∫ ay=0= 1 2 a2x=0(∫ yy dydAdx dydxx=0)dyNote that in this example, because the upper end of the x valuesdepends on the value of y, it makes a difference which order wedo the integrals in. The x integral has to be on the inside, and wehave to do it first.Volume of a cube example 19⊲ Find the volume of a cube with sides of length a.⊲ This is a three-dimensional example, so we’ll have integralsnested three deep, and the thing we’re integrating is the volumedV = dx dy dz.272 Chapter 4 Conservation of Angular Momentum