Calculus Workbook: The Fundamental Theorem of Calculus

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Section 4.3 The Fundamental Theorem of Calculus

Objectives

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Evaluate and determine basic characteristics of accumulation functions
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State the Fundamental Theorem of Calculus (Part 1) and use it to compute derivatives of accumulation functions
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State the Fundamental Theorem of Calculus (Part 2) and use it to compute definite integrals of functions with known antiderivatives (possibly with algebraic/trigonometric manipulation)
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Explain the relationship between integration and differentiation as processes

Subsection 4.3.1 Before Class

https://mymedia.ou.edu/media/4.3-1/1_9m7tun82

Figure 41. Pre-Class Video 1

Subsubsection 4.3.1.1 The First Fundamental Theorem

Example 4.3.1.

The function

f

is given in the graph below. Define the function

g (x) = \int_{0}^{x} f (t) d t .

Find

g (0), g (1), g (2),

and

g (3) .

Then, sketch a rough graph of

g

on the interval

[0, 3] .

$The graph of an accumulation function, a line going from \((0,0)\) to \((1,2)\text{,}\) a horizontal line from \((1,2)\) to \((2,2)\text{,}\) and \(2\cos\lrpar{\dfrac{\pi}{2}(x-2)}\) from \((2,2)\) to \((5,-2)\text{.}\)$

\begin{aligned} g (0) & = 0 \\ g (1) & = 1 \\ g (2) & = 3 \\ g (3) & = 3 \end{aligned}

Answers for the sketch will vary

Theorem 4.3.2. The Fundamental Theorem of Calculus, Part 1 (FTC1).

If

f

is continuous on

[a, b],

then the function

g

defined by

g (x) = \int_{a}^{x} f (t) d t,

for

a \leq x \leq b

is continuous on

[a, b],

differentiable on

(a, b),

and

g^{'} (x) = f (x) .

Using Leibniz notation, we have

\frac{d}{d x} [\int_{a}^{x} f (t) d t] = f (x)

Example 4.3.3.

Find the derivative of the function

g (x) = \int_{0}^{x} \sqrt{1 - t^{2}} d t

g^{'} (x) = \sqrt{1 - x^{2}}

Example 4.3.4.

Find

\frac{d}{d x} \int_{1}^{x^{2}} \csc t d t

\csc (x^{2}) \cdot 2 x

Example 4.3.5.

Find

F^{'} (x),

if

F (x) = \int_{x}^{0} \sqrt{1 + \sec y} d y

F^{'} (x) = - \sqrt{1 + \sec x}

Subsection 4.3.2 Pre-Class Activities

Example 4.3.6.

Let

g (x) = \int_{0}^{x} f (t) d t,

where

f

is the function in the graph below.

🔗
Evaluate $g (0), g (1), g (2), g (3),$ and $g (6)$
🔗
Where is $g$ increasing?
🔗
Where does $g$ have a local max?
🔗
Sketch a rough graph of $g ?$

$g (0) = 0,$ $g (1) = 2,$ $g (2) = 3,$ $g (3) = 5,$ $g (6) = 1$
$(0, 3)$
$t = 3$
Answers vary.

Example 4.3.7.

Find the derivative of the following functions:

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$g (x) = \int_{2}^{x} \sqrt{k + k^{3}} d k$
🔗
$r (y) = \int_{y}^{2} t^{3} \cos t d t$
🔗
$f (x) = \int_{0}^{x^{4}} \tan^{2} t d t$

$g^{'} (x) = \sqrt{x + x^{3}}$
$r^{'} (y) = - y^{3} \cos y$
$f^{'} (x) = 4 x^{3} \tan^{2} (x^{4})$

Example 4.3.8.

If

f (x) = \int_{0}^{x} (1 - t^{2}) \cos^{2} t d t,

on what interval is

f

increasing?

0 < x < 1

Subsection 4.3.3 In Class

Subsubsection 4.3.3.1 The Second Fundamental Theorem

Theorem 4.3.9. The Fundamental Theorem of Calculus, Part 2 (FTC2).

If

f

is continuous on

[a, b],

then

\int_{a}^{b} f (x) d x = F (b) - F (a),

where

F

is any antiderivative of

f .

Example 4.3.10.

Evaluate

\int_{- 2}^{1} x^{3} d x

\int_{- 2}^{1} x^{3} d x = - \frac{15}{4}

Example 4.3.11.

Find the area under the curve

y = x^{2}

from 0 to 1.

\int_{0}^{1} x^{2} d x = \frac{1}{3}

Example 4.3.12.

Find the area under the curve

y = \sin x,

from

0

to

\frac{3 π}{2}

\int_{0}^{3 π / 2} \sin x d x = 1

Example 4.3.13.

Is the statement

\int_{- 1}^{1} \frac{1}{x^{3}} d x = 0

correct? Why or why not?

No-

\frac{1}{x^{3}}

is not continuous at

x = 0,

so we can’t use FTC 2.

Example 4.3.14.

Find the derivative of

g (r) = \int_{5}^{r} (t - t^{2})^{8} d t

g^{'} (r) = (r - r^{2})^{8}

Example 4.3.15.

Find the derivative of

R (y) = \int_{y}^{4} t^{5} \sec t d t

R^{'} (y) = - y^{5} \sec y

Example 4.3.16.

Find the derivative of

y = \int_{0}^{4 x^{3}} \tan^{2} θ d θ

\frac{d y}{d x} = 12 x^{2} \tan^{2} (4 x^{3})

Example 4.3.17.

Calculate the integral

\int_{1}^{3} (x^{2} + 2 x - 4) d x

\int_{1}^{3} (x^{2} + 2 x - 4) d x = \frac{26}{3}

Example 4.3.18.

Calculate the integral

\int_{0}^{2} (\frac{4}{5} t^{3} - \frac{3}{4} t^{2} + \frac{2}{5} t) d t

\int_{0}^{2} (\frac{4}{5} t^{3} - \frac{3}{4} t^{2} + \frac{2}{5} t) d t = 2

Example 4.3.19.

Calculate the integral

\int_{1}^{9} \sqrt{x} d x

\int_{1}^{9} \sqrt{x} d x = \frac{52}{3}

Example 4.3.20.

Calculate the integral

\int_{1}^{4} \frac{2 + x^{2}}{\sqrt{x}} d x

\int_{1}^{4} \frac{2 + x^{2}}{\sqrt{x}} d x = \frac{82}{5}

Example 4.3.21.

Calculate the integral

\int_{π / 6}^{π / 2} \csc t \cot t d t

\int_{π / 6}^{π / 2} \csc t \cot t d t = 1

Example 4.3.22.

Calculate the integral

\int_{- 1}^{2} (3 u - 2) (u + 1) d u

\int_{- 1}^{2} (3 u - 2) (u + 1) d u = \frac{9}{2}

Example 4.3.23.

Calculate the integral

\int_{π / 4}^{π / 3} \csc^{2} θ d θ

\int_{π / 4}^{π / 3} \csc^{2} θ d θ = 1 - \frac{1}{\sqrt{3}}

Example 4.3.24.

Calculate the integral

\int_{1}^{18} \sqrt{\frac{2}{z}} d z

\int_{1}^{18} \sqrt{\frac{2}{z}} d z = 12 - 2 \sqrt{2}

Example 4.3.25.

Evaluate the integral

\int_{1}^{2} \frac{v^{5} + 3 v^{6}}{v^{4}} d v

\int_{1}^{2} \frac{v^{5} + 3 v^{6}}{v^{4}} d v = \frac{17}{2}

Example 4.3.26.

Evaluate the integral

\int_{0}^{π} f (x) d x,

where

f (x) = {\begin{cases} \sin x & if 0 \leq x < π / 2 \\ \cos x & if π / 2 \leq x \leq π \end{cases}

\int_{0}^{π} f (x) d x = 0

Example 4.3.27.

Evaluate the limit

lim_{n \to \infty} \sum_{i = 1}^{n} (\frac{i^{4}}{n^{5}} + \frac{i}{n^{2}}) .

Hint: Consider it as a Riemann sum on

[0, 1] .

lim_{n \to \infty} \sum_{i = 1}^{n} (\frac{i^{4}}{n^{5}} + \frac{i}{n^{2}}) = \int_{0}^{1} x^{3} + x d x = \frac{7}{10}

Example 4.3.28.

If

f (1) = 12,

f^{'}

is continuous, and

\int_{1}^{4} f^{'} (x) d x = 17,

what is the value of

f (4) ?

f (4) = 29

Subsection 4.3.4 After Class Activities

Example 4.3.29.

Compute

\int_{0}^{1} (1 + r)^{3} d r

\int_{0}^{1} (1 + r)^{3} d r = \frac{7}{4}

Example 4.3.30.

Find the area under the curve

f (x) = \frac{x^{4} + 1}{x^{2}}

between 1 and 2.

\int_{1}^{2} f (x) d x = \frac{17}{6}

Example 4.3.31.

Evaluate

\int_{- 1}^{1} x^{2022} d x

\int_{- 1}^{1} x^{2022} d x = \frac{2}{2023}

Example 4.3.32.

What does it mean for differentiation and integration to be inverse processes?

Differentiation and integration undo each other; you take a deriative to undo an integral, and you take an integral to undo a derivative.

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