Mathematical Methods for Physics 1, Solutions to Workshop Sheet 5
1
Mathematical Methods for Physics 1, Autumn 2012 – Department of Mathematics, University
of Sussex
Solutions to Workshop Sheet 5
1.
Find the
area under the graph
of the function
f
(
x
) =
1
x
,
where
x
n
= 0
,
between (a)
x
= 1 and
x
= 2, (b)
x
=
−
2 and
x
=
−
1.
(c) Sketch the graph of
f
(
x
) = 1
/x
and indicate the areas under the graph from (a) and (b).
What is the relation between the answer from (a) and (b)? Explain why this can be seen from
your sketch.
Solution:
(a) The areas is
i
2
1
f
(
x
)
dx
=
i
2
1
1
x
dx
=
(
ln

x

)v
v
2
1
= ln(2)
−
ln(1) = ln(2)
.
(b) The area is
i
−
1
−
2
f
(
x
)
dx
=
i
−
1
−
2
1
x
dx
=
(
ln

x

)v
v
−
1
−
2
= ln(1)
−
ln(2) =
−
ln(2)
.
(c) We sketch the graph of
f
(
x
) = 1
/x
and indicate the areas from (a) and (b).
2
1
1
2
1
2
1
2
Figure 1: Graph of
f
(
x
) = 1
/x
and the area from
x
= 1 to
x
= 2 and the area from
x
=
−
2 to
x
=
−
1.
We observe that
i
2
1
1
x
dx
= ln(2) =
−
i
−
1
−
2
1
x
dx.
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Mathematical Methods for Physics 1, Solutions to Workshop Sheet 5
This is to be expected from the graph of the
odd
function
f
(
x
) = 1
/x
for the following reasons:
Once we have indicated the areas under the graph from
x
= 1 to
x
= 2 and from
x
=
−
2 to
x
=
−
1, we see that both areas are the same (because
f
(
x
) = 1
/x
is odd, see also Problem 2),
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 Fall '14
 Physics, Work, dx, Department of Mathematics

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