Problems. Integral calculus
1.
Evaluate the indefinite integral:
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2.
Evaluate the indefinite integral:
3.
Evaluate the indefinite integral:
4.
Evaluate the indefinite integral:
.5.
Evaluate the indefinite integral:
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6.
Evaluate the indefinite integral:
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7.
Evaluate the indefinite integral:
8.
Evaluate the indefinite integral:
9.
Evaluate the indefinite integral:
10.
Evaluate the indefinite integral:
11.
Evaluate the indefinite integral:
12.
Evaluate the indefinite integral:
13.
Evaluate the indefinite integral:
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14.
Evaluate the indefinite integral:
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15.
Evaluate the indefinite integral:
16.
Evaluate the indefinite integral:
17.
Evaluate the indefinite integral by per partes method:
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18.
Evaluate the indefinite integral by per partes method:
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19.
Evaluate the indefinite integral:
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20.
Evaluate the indefinite integral:
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21.
Evaluate the indefinite integral:
22.
Evaluate the definite integral:
23.
Evaluate the definite integral:
24.
Evaluate the definite integral:
25.
Evaluate the definite integral:
26.
Evaluate the definite integral:
27.
Evaluate the definite integral:
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28.
Calculate the area of region bounded by the curves:
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29.
Calculate the area of region bounded by the curve:
30.
Find the volume generated by revolving the area bounded by the parabola:
about the -axis for
.
31.
Find the volume generated by revolving the first- and second-quadrant area bounded by
the ellipse:
about the -axis.
32.
Find the length of the arc of the curve:
33.
Find the length of the arc of the curve:
34.
Find the area of the surface of revolution generated by revolving about -axis the arc of
the ellipse:
.
and
and -axis
from
to
for
.
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Solutions
1.
2.
3.
4.
5.
6.
7.
Absolute value can be omitted since
8.
9.
10.
11.
for all
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12.
13.
14.
15.
16.
17.
We obtain an equation:
from which:
18.
=
19.
20.
21.
First we transform the rational function to partial fractions:
For the coefficients:
Second integral can be evaluated by a recursive formula:
We obtain:
22.
23.
24.
25.
26.
27.
28.
Surface area of the region bounded by the curves:
can be obtained as definite integral:
First we find the crossing points of the curves:
and
.
Fig. 1.
29.
Region bounded by the curves:
and
.
Surface area of the region bounded by the symmetric curve:
can be calculated as 4 times the region located in the first quadrant:
Fig. 2. Region bounded by the curve:
30.
Volume generated by revolving the area bounded by the parabola:
-axis for
can be calculated as:
about the
Fig. 3. Solid of revolution generated by revolving the area bounded by the parabola:
about -axis on the interval
.
31.
Volume generated by revolving the first- and second-quadrant area bounded by the
ellipse:
about the -axis can be evaluated as definite integral for the
boundaries of integration
:
y
Fig. 4. Solid of revolution generated by revolving the first- and second-quadrant area bounded by the
ellipse:
about -axis.
32.
The length of the arc of the curve:
calculated as:
First we calculate:
on the interval:
.
can be
33.
The length of the arc of the curve:
on the interval
will
be calculated as:
Fig. 5. Graph of a curve:
34.
The area of the surface of revolution generated by revolving about -axis the arc of the
ellipse:
with the length of its main axis equal to 4 will be calculated as:
.
First we evaluate:
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