Math 210 Lecture 11: Integration in polar
coordinates
Daniel Ingebretson
University of Illinois at Chicago
July 5, 2024
,Illustration: Area under a curve and volume under a
surface
√
Write an integral for the area under f (x) = 3 x on the
interval [4, 16].
Express the volume below the surface f (x, y ) = 2 + y1
and above the region R in the xy -plane bounded by the
lines y = x, y = 8 − x, and y = 1 as an iterated integral.
,Example: area between two curves
Find the area between f (x) = 5 − x 2 and g(x) = x 2 − 3.
, Volume between two surfaces
coordinates
Daniel Ingebretson
University of Illinois at Chicago
July 5, 2024
,Illustration: Area under a curve and volume under a
surface
√
Write an integral for the area under f (x) = 3 x on the
interval [4, 16].
Express the volume below the surface f (x, y ) = 2 + y1
and above the region R in the xy -plane bounded by the
lines y = x, y = 8 − x, and y = 1 as an iterated integral.
,Example: area between two curves
Find the area between f (x) = 5 − x 2 and g(x) = x 2 − 3.
, Volume between two surfaces
