Math 31A
2010.02.23
MATH 31A DISCUSSION
JED YANG
1.
Basic Integration
1.1.
Exercise 5.1.71.
Describe the area represented by the limit
lim
N
→∞
3
N
N
summationdisplay
j
=1
parenleftbigg
2 +
3
j
N
parenrightbigg
4
.
Solution.
Consider the area under the curve
f
(
x
) from
x
=
a
to
x
=
b
. We divide
the interval
b
−
a
into
N
equal slots, so each slot has width Δ
x
=
b

a
N
. The right
end points of the
j
th rectangle is
a
+
j
Δ
x
. The value of the function at the right
end point of the
j
th rectangle is therefore
f
(
a
+
j
Δ
x
). If we use these as the height
of the rectangles, then the area of the
j
th rectangle is Δ
x
·
f
(
a
+
j
Δ
x
). We sum
this up to get
∑
N
j
=1
Δ
x
·
f
(
a
+
j
Δ
x
). We can factor out the width of the rectangles
as they are all the same. Thus we get Δ
x
∑
N
j
=1
f
(
a
+
j
Δ
x
). If we let
N
goes to
infinity, the approximation becomes finer and finer, and approaches the area under
the curve.
Here
a
= 2,
b
−
a
= 3 so
b
= 5, and
f
(
x
) =
x
4
. Thus the limit represents the area
between the graph of
x
4
and the
x
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '07
 JonathanRogawski
 Math, perpendicular bisector, JED YANG, th rectangle

Click to edit the document details