intgrat3 ============================================== Purpose ---------------- Integrates the following triple integral, using user-defined functions and scalars for bounds: .. math:: \int_{a}^{b}\int_{g_2(x)}^{g_1(x)} \int_{gh_2(x, y)}^{h_1(x, y)}f(x, y, z)\:dz\:dy\:dx Format ---------------- .. function:: y = intgrat3(&f, xl, gl, hl) :param &f: pointer to the procedure containing the function to be integrated. *f* is a function of *(x, y, z)*. :type &f: scalar :param xl: the limits of *x*. These must be scalar limits. :type xl: 2x1 or 2xN matrix :param gl: function pointers. These procedures are functions of *x*. :type gl: 2x1 or 2xN matrix :param hl: function pointers. These procedures are functions of *x* and *y*. :type hl: 2x1 or 2xN matrix :return y: estimated integral(s) of :math:f(x, y, z) evaluated between the limits given by *xl*, *gl* and *hl*. :rtype y: Nx1 vector Global Input ------------ +-----------------+-----------------------------------------------------+ | \_intord | scalar, the order of the integration. The larger | | | \_intord, the more precise the final result will | | | be. \_intord may be set to 2, 3, 4, 6, 8, 12, 16, | | | 20, 24, 32, 40. | | | Default = 12. | +-----------------+-----------------------------------------------------+ Examples ---------------- :: // Function to be integrated proc f(x, y, z); retp(2); endp; /* ** Function of x ** to be integrated */ proc g1(x); retp(sqrt(25 - x^2)); endp; /* ** Function of x ** to be integrated */ proc g2(x); retp(-g1(x)); endp; /* ** Function of x and y ** to be integrated */ proc h1(x, y); retp(sqrt(25 - x^2 - y^2)); endp; /* ** Function of x and y ** to be integrated */ proc h2(x, y); retp(-h1(x, y)); endp; // Limits of x xl = 5|-5; // Create pointer vectors g0 = &g1|&g2; h0 = &h1|&h2; // The order of the integration _intord = 40; // Find integral y = intgrat3(&f, xl, g0, h0); This will integrate the function :math:f(x, y, z) over the sphere of radius 5. The result will be approximately twice the volume of a sphere of radius 5. Remarks ------- User-defined functions *f*, and those used in *gl* and *hl* must either: #. Return a scalar constant       - or - #. Return a vector of function values. :func:intgrat3 will pass to user-defined functions a vector or matrix for *x* and *y* and expect a vector or matrix to be returned. Use .* and ./ operators instead of just * and /. Source ------ intgrat.src Globals ------------ *_intord*, *_intq12*, *_intq16*, *_intq2*, *_intq20*, *_intq24*, *_intq3*, *_intq32*, *_intq4*, *_intq40*, *_intq6*, *_intq8* .. seealso:: Functions :func:intgrat2, :func:intquad1, :func:intquad2, :func:intquad3, :func:intsimp