Posted: April 13th, 2023

Calculus Help

Name: The Definite Integral Section:
5.2 The Definite Integral
Vocabulary Examples
Definite Integral
For a function f defined on the interval [a, b],
R
b
a
f(x) =
Average Value
(of a function) For a function f on the interval [a, b]
fave =
Properties of Definite Integrals
R
a
a
f(x)dx =
R
b
a
f(x)dx =
R
b
a
(f(x) + g(x)) dx =
R
b
a
(f(x) − g(x)) dx =
R
b
a
c · f(x)dx =
R
b
a
f(x)dx =
1. Given that R
1
0
x =
1
2
,
R
1
0
x
2 =
1
3
, and R
1
0
x
3 =
1
4
, determine the following definite integrals.
(a) R
1
0
(1 + x + x
2 + x
3
)dx (b) R
1
0
(1 − x + x
2 − x
3
)dx (c) R
1
0
(6x −
4
3
x
2
)dx
(d) R
1
0
(1 − x)
2dx (e) R
1
0
(1 − 2x)
3dx (f) R
1
0
(7 − 5x
3
)dx
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 70
Name: The Definite Integral Section:
For each definite integral R
b
a
f(x)dx, sketch a graph each function f over the interval [a, b] and then
use that graph to evaluate the definite integral.
(a) R
4
0
(4 − |x − 4|)dx (b) R
2
−2
(
√
4 − x
2)dx
R
0
−5
(
√
25 − x
2)dx R
4
0

2
3
|x − 3| − 1

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dx
2. The graph of f is shown below. Evaluate R
8
2
f(x)dx
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 71
Name: The Fundamental Theorem of Calculus Section:
5.3 The Fundamental Theorem of Calculus
Vocabulary Examples
Mean Value
Theorem of
Integrals
If f is continuous over the interval [a, b], then there exists
at least one value c [a, b] such that
f(c) =
Furthermore, it can also be said that
R
b
a
f(x)dx =
The
Fundamental
Theorem of
Calculus
If f is continuous over the interval [a, b] and
F(x) =
R
x
a
f(t)dt, then
F
0
(x) =
Furthermore, if F(x) is any anti-derivate of f(x), then
R
b
a
f(x)dx =
1. Restate the second part of the Fundamental Theorem of Calculus in your own words.
Evaluate the following integrals.
2. R
x
0
3dt 3. R
x
0
tdt 4. R
x
0
1
3
t
2dt
5. R
x
0
dt 6. R
x
0
(2 − 6t
2
)dt 7. R
x
0
(1 + cost) dt
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 72
Name: The Fundamental Theorem of Calculus Section:
Evaluate the following integrals.
8. R
2
0
3dx 9. R
4
1
x
−1dx 10. R
2
−2
e
xdx
11. R π/2
π/3
(sin x)dx 12. R
2
−2
7x
3dx 13. R
2
1

1+x
2−3x
3
x
5

dx
14. The figure below shows the graphs of f(x) = 12x − 6 − 4x
2
and g(x) = x
3 − 3x
2 + 2x + 2.
Determine the area of the shaded region.
1 2 3
1
2
3
4
15. Determine the the area bound by f(x) =
√
x and g(x) = x
2
on the interval [0, 1]
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 73
Name: Differentiation Rules Practice 2 Section:
A.4 Differentiation Rules Practice 2
1. Determine the equation of the line tangent to the function f(x) =
x
2−4
(x−2)2
at x = 3
2. The height of a particle can be modeled by the function h(t) = 4.2t − 0.4t
4
(a) Sketch a reasonable graph of the function. Be sure to consider all critical points.
(b) Based on your sketch, what is the derivative of h at the projectile’s highest point?
(c) In this real-world context, what does h
0
(t) represent?
(d) Determine the time at which the projectile reaches it’s greatest height.
(e) Determine the maximum height of the projectile.
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 95
Name: Differentiation Rules Practice 2 Section:
3. Determine the equation of a line that is tangent to the function f(x) = 25 − x
2
and also contains the
point (13, 0).
4. The population, in millions, of arctic flounder in the Atlantic Ocean is modeled by the function P(t) =
8t+3
0.2t
2+1
(a) Determine the initial population of arctic flounder.
(b) Determine P
0
(10). What does this mean in the context of the arctic flounder?
5. Given the function f(x) = −
x
2
4
+ 8.5x − 60.69, solve the following equations.
(a) f(x) = 0
(b) f
0
(x) = 2
(c) f
0
(x) = −2
(d) f
0
(x) = 0
(e) How does our answer to part (d) relate to the original function f ?
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 96
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