Iterated Integrals

The iterated integral

$\displaystyle \int_a^b\left[\int_c^d f(x,y) dy\right] dx$

is calculated by applying integrate() twice.

--> integrate(integrate(f(x,y),y,c,d),x,a,b)

The applications include (1) double integrals over general regions of type I and II, and (2) double integrals in polar coordinates.

The integral

$\displaystyle \int_{-1}^1\left[\int_{2x^2}^{x^2+1} (x+2y) dy\right] dx$

is carried out by

--> f(x,y) := x+2*y;
--> h(x) := 2*x^2;
--> k(x) := x^2 + 1;
--> integrate(integrate(f(x,y),y,h(x),k(x)),x,-1,1)

The integral

$\displaystyle \int\!\!\int_{R} (3x+4y^2) dx dy$

over $R = \{(r\cos\theta, r\sin\theta): 1 \le r \le 2, 0 \le \theta \le \pi\}$ is carried out by

--> f(x,y) := 3*x+4*y^2;
--> integrate(integrate(f(r*cos(s),r*sin(s))*r,r,1,2),s,0,%pi)

The computation of Jacobian for

and

is carried out by

--> [x,y]: [u^2-v^2, 2*u*v]
--> determinant(jacobian([x,y],[u,v]))

The triple integral

$\displaystyle \int_r^s\!\int_c^d\!\int_a^b f(x,y,z) dx dy dz$

is similarly calculated by applying integrate() recursively.

--> integrate(integrate(integrate(f(x,y,z),x,a,b),y,c,d),z,r,s)

Here are some of the even-numbered exercises assigned in the class.