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Title: Integral calculus problem with solutions.
Description: Integral calculus practice problem with a solution. it is not a handwritten pdf so it gives a clear explanation .
Description: Integral calculus practice problem with a solution. it is not a handwritten pdf so it gives a clear explanation .
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ALGEBRA
Integral Calculus
Exercises
6
...
The Indefinite Integral
In problems 1 through 7, find the indicated integral
...
R √xdx
Solution
...
3
...
ex
5
...
Solution
...
8
...
Solution
...
Thus
and so f(x) is the indefinite integral
Using the fact that the graph of f passes through the point (1,3) you get
or
...
9
...
6t2 + 0
...
5 thousand
people per year
...
By how much will the pollution in the lake increase during the next 2 years?
Solution
...
Then the rate of change of the population concerning time is the
derivative
...
6t2 + 0
...
5
...
6t2 + 0
...
5)dt =
=
0
...
1t2 + 0
...
During the next 2 years, the population will grow on
behalf of
P(2)
C=
Hence, the pollution in the lake will increase on behalf of 5 · 3 = 15 units
...
An object is moving so that its speed after t minutes is v(t) = 1+4t+3t2
meters per minute
...
Let s(t) denote the displacement of the car after t minutes
...
During the 3rd minute, the object travels
s(3)
C=
Homework
In problems 1 through 13, find the indicated integral
...
¢
14
...
15
...
2 for each value of
x and whose graph passes through the point (0,6)
16
...
17
...
If the current
population is 10000, what will the population be 8 months from now?
18
...
1t + 0
...
If the current level of carbon
monoxide in the air is 3
...
After its brakes are applied, a certain car decelerates at the constant rate
of 6 meters per second squared
...
)
20
...
If it is traveling at 90-kilo meters per hour (25 meters per
second) when the brakes are applied, its stopping distance is 50 meters
...
; not unique
18
...
15 parts per million
19
...
(a) A = 6
...
37 kilometers per hour
6
...
1
...
Substituting u = 2x + 6 and 12du = dx, you get
...
,
you get
...
3
...
4
...
5
...
6
...
√x4−x2+6 Substituting u = x4 −x2 +6 and 25du = (10x3 −5x)dx, you get
7
...
8
...
9
...
Solution
...
Use an appropriate change of variables to find the integral
Solution
...
1
...
24
...
25
...
26
...
After two years it has reached a height of five meters
...
It is projected that t years
from now the population of a certain country will be changing at the rate
of e0
...
If the current population is 50 million, what will
the population be 10 years from now?
Results
...
61 million
6
...
1
...
1xdx
Solution
...
1x is easy to integrate and the factor x is
simplified by differentiation, try integration by parts with
g(x) = e0
...
Then,
G(x) = Z e0
...
1x and f0(x) = 1 and so
2
...
is simplified by differentiation, try integration by parts with g(x)
= e−x and f(x) = 3 − 2x
...
3
...
In this case, the factor x is easy to integrate, while the factor lnx2
is simplified by differentiation
...
and
Then,
and
and so
4
...
R √1 − Since the
factoris easy to integrate and the factor
x is simplified by differentiation, try integration by parts with
and
f(x) = x
...
5
...
Since the factor (x + 2)6 is easy to integrate and the factor
x + 1 is simplified by differentiation, try integration by parts with g(x) = (x
+ 2)6 and f(x) = x + 1
...
R x3e2xdx
Solution
...
f0(x) = 3x2
and so
...
f0(x) = 2x
...
f0(x) = 1
...
Solution
...
This suggests that you try integration by
parts with
and
f(x) = lynx
...
R x3ex2dx
³
2
2
´
Solution
...
Then, from Exercise 6
...
3 you get
and
f0(x) = 2x
and so
9
...
R x3(x2 − 1)10First rewrite the integrand asdx x2[x(x2 −
1)10], and then
g(x) = x(x2 − 1)10
and
f(x) = x2
...
...
(b) Use the formula in part (a) to find R x3e5xdx
...
(a) Since the factor eax is easy to integrate and the factor xn
is simplified by differentiation, try integration by parts with g(x) = eax
and f(x) = xn
...
(b) Apply the formula in part (a) with a = 5 and n = 3 to get
...
Once again, apply the formula in part (a) with a = 5 and n = 1 to get
and so
Homework
In problems 1 through 16, use integration by parts to find the given integral
...
17
...
18
...
19
...
Express the distance the object travels as a function of time
...
It is projected that t years from now the population of a certain city will be
changing at the rate of tln√t + 1 thousand people per year
...
1
...
6
...
1
...
First rewrite the integrand as
and
then
substitute
2
...
First, rewrite the integrand as
and then substitute u = x + 1 and du = dx to get
3
...
First, rewrite the integrand as
...
4
...
Since the expression
...
Solution
...
Homework
In Problems 1 through 10, use one of the integration formulas listed in this
section to find the given integral
...
3
...
7
...
Locate a table of integrals and use it to find the integrals in Problems 11
through 16
...
One table of integrals lists the formula
while another table lists
...
The following two formulas appear in a table of integrals: ¯¯ ¯¯
and
(a) Use the second formula to derive the
(b) Apply both formulas to the integral
to use in this problem?
...
3
...
5
The Definite Integral
In problems, 1 through 7 evaluate the given definite integral
1
...
2
...
(0) = 6
...
Then 12du = dx, u(−3) = 0,
and
...
+1)
Solution
...
Then 13du = x2dx, u(1) = 2, and u(2) = 9
...
4
...
Hence,
...
Solution
...
and
and
f(t) = ln2t
1(
twice to get
7
...
Integrate by parts with
and
Then,
f(t) = t
and
f0(t) = 1
and so
(a) Show that
...
(c) Evaluate
...
(a) By the Newton-Leibniz formula, you have
Z bc
Z
f(x)dx
+f(x)dx
=
F(b) − F(a) + F(c) − F(b)
= ab
=
F(c) − F(a) = Za f(x)dx
...
(a) Show that if F is
Z
an antiderivative of f, then
b
f(−x)dx = −F(−b) + F(−a)
a
(b) A function f is said to be even if f(−x) = f(x)
...
] Use problem 8 and part (a) to show that if is even, then
(c) Use part (b) to evaluate
...
Use problem 8 f is odd, then
...
u(b) =−(−a) Substituteb
...
Then du
= −dx, u(a) = −a
...
By
Z
the part (a), you have
0
f(−x)dx
= −F(0) + F(−(−a)) = F(a) − F(0) =
−a
=Z
a
f(x)dx
...
(c) Since f(x) = |x| is an even function, you have
...
(e) Since f(x) = x3 is an
Z
odd function, you have
12
x3dx = 0
...
It is estimated that t days from now a farmer’s crop will be increasing at the
rate of 0
...
6t + 1 bushels per day
...
Let Q(t) denote the farmer’s crop t days from now
...
Homework
In problems, 1
through
17,
evaluate
the
given definite
integral
1
...
18
...
By how much will
the population of the town increase over the next 8 months?
19
...
If the oil demand is currently 30 billion barrels per
year, how much oil will be consumed during the next 10 years?
20
...
How far does the object travel during the 2nd minute?
Results
...
3
...
7
...
billion barrels
20
...
6
Area and Integration
In problems, 1 through 9 find the area of the region R
...
resolution
...
1) you see that the(−4,0), (2,0,) and (2,6)
...
y
6
4
y=x+4
-4
2
-2
2
0
x
Figure 6
...
Hence,
2
2
16) = 18
...
and x = ln 12, and the x axis
...
2) you see that the region in question
isSolution
...
7, from the corbellow the line
y = ex above the x axis, and extends from x = ln 12 to x = 0
...
2
...
3
...
Solution
...
3
...
Also, note that
the line
y = −x + 10 intersects the axis at the point (10,0)
...
3
...
This
suggests that you break R into two subregions, R1 and R2, as shown in
Figure 6
...
In particular,
and
...
4
...
Solution
...
4
...
4
...
Hence,
¯
...
Solution
...
5 and find
the points of intersection of the two curves by solving the equation
x2 − 2x = −x2 + 4
i
...
2x2 − 2x − 4 = 0
x = −1
and
x = 2
...
3
Figure 6
...
Notice that for −1 ≤ x ≤ 2, the graph of y = −x2 + 4 lies above that of y = x2
− 2x
...
R is the region bounded by the curves
...
Sketch the region as shown in Figure 6
...
Find the points of
intersection by solving the equations of the two curves simultaneously to
get
−
− x = 0 and x = 1
...
Figure 6
...
Notice that for2 0 ≤ x ≤ 1, the graph lies above that of y = x
...
(a) R is the region to the right of the y axis that is bounded above by the
curve y = 4 − x2 and below the line y = 3
...
Solution
...
7
...
e
...
Figure 6
...
ofNotice that for 0 ≤ x ≤ 1, the graph of y = 4 −x2 lies above that
y = 3
...
Figure 6
...
Observe that to the left of x = 1, R is bounded above by the curve y =
3, while to the right of x = 1, it is bounded by the line y = 4 −Rx22, as
shown in Figure 6
...
This suggests
that you break R into two subregions,
R1 and
for the area to each subregion separately
...
R
is the region bounded by the curve
and the lines y = x and y 8
...
First, sketch the region as shown in Figure 6
...
e
...
Figure 6
...
and
x3 = 8
Then break R into two subregions, R1 that extends from x = 0 to x = 1 and
R2 that extends from x = 1 to x = 2, as shown in Figure
6
...
Hence, the area of the region R1 is
and the area of the region R2 is
...
8
...
2 +
4x − 7
...
10
...
Figure 6
...
The region whose area you wish to compute lies between x = −2 and x =
3, but since the two curves cross at x = 2, neither curve is always above the
other between x = −x22 +4andxx−== 37xbetween3
...
Homework
In problems, 1 through 20 find the area of the region R
...
R is the triangle bounded by the line y = 4 − 3x and the coordinate axes
...
R is the rectangle with vertices (1,0), (−2,0), (−2,5), and (1,5)
...
R is the trapezoid bounded by the lines y = x + 6 and x = 2 and the
coordinate axes
...
R is the region bounded by the curve
, the line x = 4, and the x-axis
...
R is the region bounded by the curve y = 4x , the line x = 2, and the x-axis
...
R is the region bounded by the curve y = 1 − x2 and the x-axis
...
R is the region bounded by the curve y = −x2 −6x−5 and the x-axis
...
R is the region in the first quadrant bounded by the curve y = 4 − x2 and
the lines y = 3x and y = 0
...
and is the region bounded by the curvey = 0
...
R is the region in the first quadrant that lies under the curve
that is bounded by this curve and the lines y = x, y = 0, and x = 8
...
ReR is the region bounded by the reflect the region across the x-axis and
integrate the corresponding = x2−2x and the x-axis
...
)
12
...
13
...
14
...
15
...
16
...
17
...
18
...
19
...
=
y = x2 − 3x + 1 and y
−
20
...
Results
...
6
Title: Integral calculus problem with solutions.
Description: Integral calculus practice problem with a solution. it is not a handwritten pdf so it gives a clear explanation .
Description: Integral calculus practice problem with a solution. it is not a handwritten pdf so it gives a clear explanation .