Part 1: Type I Integrals

In this section, we begin the study of integrals over regions in the plane. To do so, however, requires that we examine the important idea of iterated integrals, in which indefinite integrals are the integrand of a definite integral.

To begin with, we define a type I iterated integral to be an integral of the form

ó õ b a  ó õ q( x) p( x)   f( x,y)dy dx

To evaluate a type I integral, we first evaluate the inner integral

ó õ q( x) p( x)   f( x,y) dy

treating x as a constant. We then evaluate the result with respect to x:

ó õ b a  ó õ q( x) p( x)   f( x,y)dy dx = ó õ b a  é ë   ó õ q( x) p( x)   f( x,y) dy  ù û  dx

EXAMPLE 1    Evaluate the type I integral

ó õ 1 0  ó õ x 0  ( xy2+1) dydx

Solution: To begin with, we integrate with respect to y:

ó õ x 0  ( xy2+1) dy = æ è x  y3 3 +y ö ø ê ê x 0  = æ è x  x3 3 +x ö ø - æ è x  03 3 +0 ö ø =  1 3 x4+x

As a result, we have

ó õ 1 0  ó õ x 0  ( xy2+1) dydx = ó õ 1 0  æ è  1 3 x4+x ö ø dx =  1 3  x4 4 +  x2 2 ê ê 1 0  =  1 12 +  1 2 =  7 12

       

Often we evaluate the innermost integral inside the integrand of the outer integral rather than writing the integrations separately.      

EXAMPLE 2    Evaluate the type I integral

ó õ 2 0  ó õ x 1  x2ydydx

Solution: We first evaluate the inner integral:

ó õ 2 0  ó õ x 1  x2ydydx = ó õ 2 0  é ë ó õ x 1  x2ydy ù û dx = ó õ 2 0  é ë x2  y2 2 ê ê x 1  ù û dx = ó õ 2 0  é ë x2  x2 2 -x2  1 2 ù û dx = ó õ 2 0  é ë  x4 2 -  x2 2 ù û dx =  28 15