Definite integral as an area
Let
y = f (x)
be a function in a single variable, defined on the interval [a, b].
Definition and Notation
The definite integral of f on [a, b], denoted
Z
b
f (x) dx,
a
represents the “signed area under the curve y = f (x)” between the
vertical lines at x = a and x = b.
Math 105 (Section 204)
Integral Calculus – Definite integrals
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Clarification of the term “signed area”
1. If f is always non-negative on [a, b], and S denotes the area of the
region bounded above by the curve y = f (x), below by the x-axis and
on the sides by the vertical lines at x = a and x = b (see accompanying
diagram), then
Z b
f (x) dx = S.
a
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
2/6
Clarification of the term “signed area”
1. If f is always non-negative on [a, b], and S denotes the area of the
region bounded above by the curve y = f (x), below by the x-axis and
on the sides by the vertical lines at x = a and x = b (see accompanying
diagram), then
Z b
f (x) dx = S.
a
2. If f is always non-positive on [a, b], and S denotes the area of the
region bounded below by the curve y = f (x), above by the x-axis and
on the sides by the vertical lines x = a and x = b, then
Z
b
f (x) dx = −S.
a
Thus an integral could be negative, even when it represents an area!
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
2/6
“Signed Area” ctd
3. Suppose that c is a point in [a, b], and the function f is non-negative
on [a, c] and non-positive on [c, b] (see accompanying diagram).
Suppose that the areas of the regions bounded by the curve y = f (x)
and the x-axis are, respectively, S1 on [a, c] and S2 on [c, b]. Then
Z
b
f (x) dx = S1 − S2 .
a
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
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Computation of some definite integrals : Example 1
Evaluate the integral
Z
3p
9 − x 2 dx,
0
interpreting it as an area.
A. 18
B. 9π
C.
D.
9π
2
9π
4
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
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Example 2
Find the value of the integral
Z
4
(4 − x) dx.
0
A. 8
B. 4
C. 2
D. 1
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
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Example 3
Find the value of the integral
Z
2π
sin x dx.
0
A. 1
B. 0
C. 2π
D. −2π
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
6/6