Evaluate the iterated integral ∫ 1 0 ∫ 1 sin (2 )
Answers
Answer:
valuate the iterated integral
∫
1
0
∫
1
x
s
i
n
(
y
2
)
d
y
d
x
by changing the order of integration.
Iterated Integration:
Iterated integration refers to the process used to integrate integrals of the form
∫
b
a
∫
g
2
(
x
)
g
1
(
x
)
d
y
d
x
or
∫
d
c
∫
h
2
(
y
)
h
1
(
y
)
d
x
d
y
Sometimes it is necessary to change the order
d
y
d
x
to
d
x
d
y
to obtain an integral that is easier to solve.
Answer and Explanation:
Let's evaluate the iterated integral
∫
1
0
∫
1
x
sin
(
y
2
)
d
y
d
x
by changing the order of integration.
Step 1. Sketch of the region of the form
d
x
d
y
.
Region bounded by the y-axis, the line y=1, and the line y=x
Region bounded by the y-axis, the line y=1, and the line y=x
Step 2. Calculation of the double integral.
The integral is given by
∫
1
0
∫
y
0
sin
(
y
2
)
d
x
d
y
=
∫
1
0
x
sin
(
y
2
)
∣
∣
y
0
d
y
=
∫
1
0
y
sin
(
y
2
)
d
y
=
−
1
2
cos
(
y
2
)
.
∣
∣
∣
1
0
=
−
1
2
cos
(
1
)
+
1
2
Therefore
∫
1
0
∫
y
0
sin
(
y
2
)
d
x
d
y
=
−
1
2
cos
(
1
)
+
1
2
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Double Integrals & Evaluation by Iterated Integrals
from
Chapter 15 / Lesson 4
686
In this lesson, we show how to evaluate a double integral using iterative integration. A special case is also presented which simplifies the calculations.
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