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AP Calculus BC
Unit 6: Integration and Accumulation of Change
6.6 Applying Properties of Definite Integrals
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What numerical value does the definite integral represent?
Signed area under a curve
The definite integral of
f
(
x
)
f(x)
f
(
x
)
from
x
=
x =
x
=
a
a
a
to
x
=
x =
x
=
b
b
b
is denoted as \int_{a}^{b} f(x) \, dx
The linearity property allows definite integrals to be distributed over addition and multiplied by
constants
.
Match the definite integral property with its description:
Additivity over intervals ↔️ Integrals can be summed over consecutive intervals
Even function ↔️ Integral from -a to a is twice the integral from 0 to a
Odd function ↔️ Integral from -a to a equals zero
How is the definite integral calculated?
Fundamental Theorem of Calculus
The linearity property of definite integrals allows us to move a constant outside the
integral
Use linearity to evaluate
∫
1
3
(
2
x
+
3
x
2
)
d
x
\int_{1}^{3} (2x + 3x^{2}) \, dx
∫
1
3
(
2
x
+
3
x
2
)
d
x
, given \int_{1}^{3} x \, dx = 4</latex> and
∫
1
3
x
2
d
x
=
\int_{1}^{3} x^{2} \, dx =
∫
1
3
x
2
d
x
=
26
3
\frac{26}{3}
3
26
:
1️⃣
∫
1
3
(
2
x
+
3
x
2
)
d
x
=
\int_{1}^{3} (2x + 3x^{2}) \, dx =
∫
1
3
(
2
x
+
3
x
2
)
d
x
=
2
∫
1
3
x
d
x
+
2\int_{1}^{3} x \, dx +
2
∫
1
3
x
d
x
+
3
∫
1
3
x
2
d
x
3\int_{1}^{3} x^{2} \, dx
3
∫
1
3
x
2
d
x
2️⃣
=
=
=
2
(
4
)
+
2(4) +
2
(
4
)
+
3
(
26
3
)
3\left(\frac{26}{3}\right)
3
(
3
26
)
3️⃣
=
=
=
8
+
8 +
8
+
26
26
26
4️⃣
=
=
=
34
34
34
What property allows us to split a definite integral into smaller parts over consecutive intervals?
Additivity
If
∫
1
2
x
2
d
x
=
\int_{1}^{2} x^{2} \, dx =
∫
1
2
x
2
d
x
=
7
3
\frac{7}{3}
3
7
and
∫
2
3
x
2
d
x
=
\int_{2}^{3} x^{2} \, dx =
∫
2
3
x
2
d
x
=
19
3
\frac{19}{3}
3
19
, then
∫
1
3
x
2
d
x
\int_{1}^{3} x^{2} \, dx
∫
1
3
x
2
d
x
equals \frac{26}{3}
The definite integral represents the signed area under a
curve
.
The definite integral represents the signed
area
The definite integral is denoted as
\int_{a}^{b} f(x) \, dx
</latex>.
The linearity property allows constants to be multiplied outside the
integral
The formula for linearity is \int_{a}^{b} (cf(x) + dg(x)) \,
dx
= c\int_{a}^{b} f(x) \, dx + d\int_{a}^{b} g(x) \, dx</latex>.
Additivity over intervals states that
∫
a
c
f
(
x
)
d
x
=
\int_{a}^{c} f(x) \, dx =
∫
a
c
f
(
x
)
d
x
=
∫
a
b
f
(
x
)
d
x
+
\int_{a}^{b} f(x) \, dx +
∫
a
b
f
(
x
)
d
x
+
∫
b
c
f
(
x
)
d
x
\int_{b}^{c} f(x) \, dx
∫
b
c
f
(
x
)
d
x
, where b is between
a
a
a
and
c
c
c
.
If
f
(
x
)
f(x)
f
(
x
)
is odd, then
∫
−
a
a
f
(
x
)
d
x
=
\int_{ - a}^{a} f(x) \, dx =
∫
−
a
a
f
(
x
)
d
x
=
0
0
0
.
The addition property of linearity states that
∫
a
b
[
f
(
x
)
+
\int_{a}^{b} [f(x) +
∫
a
b
[
f
(
x
)
+
g
(
x
)
]
d
x
=
g(x)] \, dx =
g
(
x
)]
d
x
=
∫
a
b
f
(
x
)
d
x
+
\int_{a}^{b} f(x) \, dx +
∫
a
b
f
(
x
)
d
x
+
∫
a
b
g
(
x
)
d
x
\int_{a}^{b} g(x) \, dx
∫
a
b
g
(
x
)
d
x
, which means the integral of a sum is the sum of the integrals.
Constant multiplication allows a constant to be moved outside the
integral
.
Match the function type with its property:
Even Function ↔️
∫
−
a
a
f
(
x
)
d
x
=
\int_{ - a}^{a} f(x) \, dx =
∫
−
a
a
f
(
x
)
d
x
=
2
∫
0
a
f
(
x
)
d
x
2 \int_{0}^{a} f(x) \, dx
2
∫
0
a
f
(
x
)
d
x
Odd Function ↔️
∫
−
a
a
f
(
x
)
d
x
=
\int_{ - a}^{a} f(x) \, dx =
∫
−
a
a
f
(
x
)
d
x
=
0
0
0
The symmetry of a function can simplify definite integrals over symmetric
intervals
What is the property of an even function in terms of its integral over a symmetric interval?
∫
−
a
a
f
(
x
)
d
x
=
\int_{ - a}^{a} f(x) \, dx =
∫
−
a
a
f
(
x
)
d
x
=
2
∫
0
a
f
(
x
)
d
x
2 \int_{0}^{a} f(x) \, dx
2
∫
0
a
f
(
x
)
d
x
What is the property of an odd function in terms of its integral over a symmetric interval?
∫
−
a
a
f
(
x
)
d
x
=
\int_{ - a}^{a} f(x) \, dx =
∫
−
a
a
f
(
x
)
d
x
=
0
0
0
Since
x
2
x^{2}
x
2
is even,
∫
−
2
2
x
2
d
x
=
\int_{ - 2}^{2} x^{2} \, dx =
∫
−
2
2
x
2
d
x
=
16
3
\frac{16}{3}
3
16
The Constant Multiple Rule states that
∫
a
b
c
f
(
x
)
d
x
=
\int_{a}^{b} c f(x) \, dx =
∫
a
b
c
f
(
x
)
d
x
=
c
∫
a
b
f
(
x
)
d
x
c \int_{a}^{b} f(x) \, dx
c
∫
a
b
f
(
x
)
d
x
What does the definite integral quantify?
Signed area under a curve
The linearity property allows constants to be multiplied into
definite integrals
.
Given
∫
1
3
x
d
x
=
\int_{1}^{3} x \, dx =
∫
1
3
x
d
x
=
4
4
4
and
∫
1
3
x
2
d
x
=
\int_{1}^{3} x^{2} \, dx =
∫
1
3
x
2
d
x
=
26
3
\frac{26}{3}
3
26
, find
∫
1
3
(
2
x
+
3
x
2
)
d
x
\int_{1}^{3} (2x + 3x^{2}) \, dx
∫
1
3
(
2
x
+
3
x
2
)
d
x
using linearity.
34
What is the formula for the additivity property of definite integrals?
\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx</latex>
The integral of an even function
from
-a to a is twice the integral from 0 to a.
What is the value of
∫
−
a
a
f
(
x
)
d
x
\int_{ - a}^{a} f(x) \, dx
∫
−
a
a
f
(
x
)
d
x
if
f
(
x
)
f(x)
f
(
x
)
is an odd function?
0
What are the two categories of functions that affect definite integrals based on symmetry?
Even and odd
The definite integral of an odd function over a symmetric interval is equal to
zero
What is the value of
∫
−
2
2
x
2
d
x
\int_{ - 2}^{2} x^{2} \, dx
∫
−
2
2
x
2
d
x
if
∫
0
2
x
2
d
x
=
\int_{0}^{2} x^{2} \, dx =
∫
0
2
x
2
d
x
=
8
3
\frac{8}{3}
3
8
?
16
3
\frac{16}{3}
3
16
The definite integral of
x
3
x^{3}
x
3
from
−
3
- 3
−
3
to
3
3
3
is
0
0
0
.
The symmetry properties of functions affect the evaluation of
definite
integrals.
What does the Constant Multiple Rule state in the context of definite integrals?
\int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx</latex>
If
∫
1
3
x
d
x
=
\int_{1}^{3} x \, dx =
∫
1
3
x
d
x
=
4
4
4
, then
∫
1
3
5
x
d
x
=
\int_{1}^{3} 5x \, dx =
∫
1
3
5
x
d
x
=
20
20
20
.
The linearity property allows distributing integrals over sums and
constants
.
Steps to evaluate
∫
0
3
x
d
x
\int_{0}^{3} x \, dx
∫
0
3
x
d
x
using additivity
1️⃣ \int_{0}^{1} x \, dx = \frac{1}{2}</latex>
2️⃣
∫
1
3
x
d
x
=
\int_{1}^{3} x \, dx =
∫
1
3
x
d
x
=
4
4
4
3️⃣
∫
0
3
x
d
x
=
\int_{0}^{3} x \, dx =
∫
0
3
x
d
x
=
1
2
+
\frac{1}{2} +
2
1
+
4
4
4
4️⃣
∫
0
3
x
d
x
=
\int_{0}^{3} x \, dx =
∫
0
3
x
d
x
=
9
2
\frac{9}{2}
2
9
If
∫
0
2
x
d
x
=
\int_{0}^{2} x \, dx =
∫
0
2
x
d
x
=
2
2
2
and
∫
0
2
x
2
d
x
=
\int_{0}^{2} x^{2} \, dx =
∫
0
2
x
2
d
x
=
8
3
\frac{8}{3}
3
8
, what is the value of
∫
0
2
(
3
x
+
2
x
2
)
d
x
\int_{0}^{2} (3x + 2x^{2}) \, dx
∫
0
2
(
3
x
+
2
x
2
)
d
x
?
34
3
\frac{34}{3}
3
34
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