3.
Double Integrals
3A.
Double Integrals in Rectangular Coordinates
3A1
Evaluate each of the following iterated integrals:
(6x2
+
2y) dy dx
b) lnI2 Ln(, sin t
+
t cos U) dt du
3A2
Express each double integral over the given region R as an iterated integral, using
the given order of integration. Use the method described in Notes I to supply the limits of
integration. For some of them, it may be necessary to break the integral up into two parts.
In each case, begin by sketching the region.
a) R is the triangle with vertices at the origin, (0,2), and (2,2).
Express as an iterated integral: i)
dy dx
ii)
/L
dx dy
b) R is the finite region between the parabola y
=
2x

x2 and the xaxis.
dy dx
ii)
/L
dx dy
c) R is the sector of the circle with center at the origin and radius 2 lying between the
xaxis and the line
u
=
x.
"
dy dx
ii)
dx dy
d)* R is the finite region lying between the parabola
y2
=
x and the line through (2,O)
having slope 1.
dy dx
ii)
dx dy
3A3
Evaluate each of the following double integrals over the indicated region R. Choose
whichever order of integration seems easier

this will be influenced both by the integrand,
and by the shape of R.
a)
x dA; R is the finite region bounded by the axes and 2y
+
x
=
2
b) /L(2x
+
Y2)
dA; R is the finite region in the first quadrant bounded by the axes
and
=
1

x; (dx dy is easier).
c)
y dA; R is the triangle with vertices at
(f
1, O), (0,l).
3A4
Find by double integration the volume of the following solids.
a) the solid lying under the graph of z
=
sin2 x and over the region R bounded below by
the xaxis and above by the central arch of the graph of cosx
b) the solid lying over the finite region R in the first quadrant between the graphs of x
and x2, and underneath the graph of z
=
xy.
.
c) the finite solid lying underneath the graph of x2

y2, above the xyplane, and between
the planes x
=
0 and x
=
1
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E. 18.02 EXERCISES
3A5
Evaluate each of the following iterated integrals, by changing the order of integration
(begin by figuring out what the region R is, and sketching it).
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 Spring '08
 Auroux
 Integrals, Polar Coordinates, Multiple integral, dx dy, Rectangular Coordinates, double integration

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