week11.html

<h3>Integration</h3>

<p>This week is an odd one; we've spent almost every week thus
far doing some differentiation, or even <em>undoing</em> differentiation as
we did most recently during <a
href="/calc1-001/wiki/view?page=Week10">Week 10</a>.  Next week, in <a
href="/calc1-001/wiki/view?page=Week12">Week 12</a>, we'll be thinking
about derivatives again when we cover the <strong>Fundamental Theorem of
Calculus</strong>.  So what is up with this week?  <a href="/calc1-001/lecture/275"><i
class="icon-film"></i>&nbsp;If we aren't differentiating, what are we
going to do?</a></p>

<p>Fundamentally, this week is about definitions, not computations.
Things are going to be a bit technical, and if this week doesn't make
perfect sense, don't fret: the rest of the course will still be
understandable.  That said, I think the material is refreshingly
different: we'll learn about summation notation and think about
the very nature of <strong>area</strong>.</p>

<p>There's also two more weeks to finish the midterm: If you have any
questions on the midterms, if you're stuck, if you're excited about
how you've solved a problem and want to share, if you just want to
vent, please <a href="/calc1-001/forum/index">post a message to the
forum</a>&mdash;I want everybody to make it through the course, and
that means we have to stay in contact with each other.  If you want to
talk in person, I'll be holding <a href="/calc1-001/wiki/view?page=OfficeHours">Office
Hours</a> on March 20, 2013 at 12:30PM EDT.</p>


<h3>What is summation notation?</h3>

<p><a
href="http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation">Sigma
notation</a> is a fancy way to write down a sum, by <a href="/calc1-001/lecture/277"><i
class="icon-film"></i>&nbsp;using a big $$\displaystyle\sum$$</a> and writing down a
pattern for each term.  For instance, if I want to add $$10 + 12 + 14
+ 16 + 18 + 20$$, I could write
<blockquote>
$$\displaystyle\sum_{n=5}^{10} \left( 2n \right).$$
</blockquote>
You can <a
href="https://mooculus.osu.edu/exercises/basicSigma"><i
class="icon-pencil"></i>&nbsp;practice summation notation on mooculus</a>.</p>

<p>Sometimes you can evaluate sums in sneaky ways.  For instance, <a href="/calc1-001/lecture/281"><i
class="icon-film"></i>&nbsp;the sum of the first $$k$$ whole
numbers</a>, called a <a
href="http://en.wikipedia.org/wiki/Triangular_number">triangular
number</a>, has a particularly nice form, namely
<blockquote>
$$\displaystyle\sum_{n=1}^{k} n = 1 + 2 + \cdots + k =
\displaystyle\frac{k \cdot (k+1)}{2}.$$
</blockquote>
Other nice patterns include the fact that <a href="/calc1-001/lecture/283"><i
class="icon-film"></i>&nbsp;the sum of the first $$k$$ odd
numbers is a perfect square</a>, which can be seen geometrically.
There's a <a href="/calc1-001/lecture/279"><i
class="icon-film"></i>&nbsp;formula for the sum of squares</a>, namely 
<blockquote>
$$\displaystyle\sum_{n=1}^{k} n^2 = 1^2 + 2^2 + 3^2 + \cdots + k^2 =
\displaystyle\frac{k \cdot (k+1) \cdot (2k+1)}{6},$$
</blockquote>
and the truly remarkable <a href="/calc1-001/lecture/285"><i
class="icon-film"></i>&nbsp;theorem of Nicomachus</a> which states 
<blockquote>
$$\displaystyle\sum_{n=1}^{k} n^3 = 1^3 + 2^3 + 3^3 + \cdots + k^3 =
\left(\displaystyle\sum_{n=1}^k n \right)^2.$$
</blockquote>
You can <a
href="https://mooculus.osu.edu/exercises/sigmaFormulas1"><i
class="icon-pencil"></i>&nbsp;practice these &ldquo;power sum&rdquo;
formulas on mooculus</a>.</p>

<p>Messing around with sums is fun, but there's a question as to why
we're bothering.  These sum formulas will end up being helpful when we
work on &ldquo;Riemann sums.&rdquo;
</p>

<h3>What is area?</h3>

<p>In our heart, we've all got an idea as to what &ldquo;area&rdquo;
means, and it's not my place to destroy your intuition!  Nevertheless,
let's ask the question: <a href="/calc1-001/lecture/293"><i
class="icon-film"></i>&nbsp;what is area?</a>  If one region can be
cut up into pieces and rearranged to form another region, then those
two regions must  &ldquo;have the same area&rdquo; but what about the
converse?  If two regions have the same area, can I demonstrate that
fact by cutting the one up into pieces and rearranging?</p>

<p>The trouble
is things like circles, what with their curved
boundaries: a circle of radius $$1$$ has area $$\pi$$, but so does a
$$1$$-by-$$\pi$$ rectangle, and yet, I can't slice that rectangle up
to get the circle.  And yet, I can chop up my rectangle into little
pieces which can be rearranged to be as close to the circle as I
want.  It's a limit!  <strong>Thinking about area leads us to
limits!</strong>  And not just us, but <a
href="http://en.wikipedia.org/wiki/Archimedes_Palimpsest">even
Archimedes</a> was lead to this sort of thinking two millenia ago.</p>

<h3>So how do we make this precise</h3>

<p>Integrals are introduced on <a href="https://mooculus.osu.edu/handouts"><i class="icon-book"></i>&nbsp;Page&nbsp;190
of the Textbook</a>.</p>

<p>We'll <a href="/calc1-001/lecture/297"><i
class="icon-film"></i>&nbsp;approximate areas</a> by forming
<strong>Riemann sums</strong>; if we want to approximate the area
under the graph $$y = f(x)$$ and between $$x = a$$ and $$x = b$$, I'll
first partition the interval $$[a,b]$$ into $$n$$ smaller pieces
$$[x_0,x_1]$$, $$[x_1,x_2]$$, $$[x_2,x_3]$$, and so forth; then I'll
pick a point $${x_i}^\star$$ in the interval $$[x_{i-1},x_i]$$ where
I'll sample the function.  My approximation to the area is then
<blockquote>
$$\displaystyle\sum_{i=1}^n f({x_i}^\star) \, \left( x_{i} - x_{i-1} \right).$$
</blockquote>
The limit of such things (over finer and finer partitions) is the
<strong>integral</strong>, which <a href="/calc1-001/lecture/299"><i
class="icon-film"></i>&nbsp;we denote $$\displaystyle\int_a^b f(x) \, dx$$.</a></p>

<p>I invite you to <a
href="https://mooculus.osu.edu/exercises/estimatingIntegrals"><i
class="icon-pencil"></i>&nbsp;gain some intuition for integration on mooculus</a>.</p>

<h3>Can we compute any of these integrals?</h3>

<p>Somewhat surprisingly, we can <a href="/calc1-001/lecture/295"><i
class="icon-film"></i>&nbsp;integrate the function $$f(x) = x^2$$</a>
by hand.  For instance, $$\displaystyle\int_0^1 x^2 \, dx = 1/3$$,
which is remarkable, since we are computing the area of a curved
region exactly.  We can also <a href="/calc1-001/lecture/303"><i
class="icon-film"></i>&nbsp;compute $$\displaystyle\int_1^2 x^3
\,dx$$</a>.  These calculations end up invoking our previous
work on summation formulas.</p>

<p>You can do some of these <a
href="https://mooculus.osu.edu/exercises/riemannSumIntegralQuadratic"><i
class="icon-pencil"></i>&nbsp;integral calculations yourself on mooculus</a>.</p>

<h3>Can we understand anything conceptually about integrals?</h3>

<p>One thing to emphasize is that <strong>integrals compute
&ldquo;signed area.&rdquo;</strong> Area, like length, cannot be
negative, but if you integrate a negative function, you should expect
a negative answer.</p>

<p>Another way to understand integrals is to fit specific integrals
together into a family of integrals; less metaphorically, I am
suggesting we study the <a href="/calc1-001/lecture/289"><i
class="icon-film"></i>&nbsp;accumulation function</a> given by $$A(x)
= \displaystyle\int_a^x f(t) \, dt$$.  By thinking about when $$A$$ is increasing
or decreasing, we get our first glimpse into the <strong>Fundamental
Theorem of Calculus</strong> coming in <a
href="/calc1-001/wiki/view?page=Week12">Week 12</a>.  You can play
around with <a
href="https://mooculus.osu.edu/exercises/integralSketch"><i
class="icon-pencil"></i>&nbsp;accumulation functions on mooculus</a>.</p>

<p>And the integral  <a href="/calc1-001/lecture/291"><i
class="icon-film"></i>&nbsp;satisfies a few &ldquo;linearity&rdquo;
properties</a>, like $$\displaystyle\int_a^b \left( f(x) + g(x) \right) \,
dx = \displaystyle\int_a^b f(x) \, dx +  \displaystyle\int_a^b g(x) \,
dx$$.</p>

<h3>Can we compute any other integrals?</h3>

<p>There is a trick that can be exploited to evaluate some integrals; if
you are <a href="/calc1-001/lecture/287"><i
class="icon-film"></i>&nbsp;integrating an odd function across the
origin</a>.  The trick is that, say, $$\displaystyle\int_{-1}^0 \sin x
\, dx = - \displaystyle\int_0^1 \sin x \, dx$$, and so
$$\displaystyle\int_{-1}^{1} \sin x \, dx = 0$$, which we have deduced
without doing any summation calculations.</p>

<h3>At the end of the week&hellip;</h3>

<p>When you've reached the end of the week, take the <a href="/calc1-001/quiz/index">End of Week 11 Quiz</a>.  So you aren't surprised when you take the quiz, you should expect</p>
<ul>
<li>three questions involving <a href="/calc1-001/lecture/277"><i class="icon-film"></i>&nbsp;summation notation</a>,</li>
<li>one question about <a href="/calc1-001/lecture/293"><i class="icon-film"></i>&nbsp;the idea of area</a>,</li>
<li>four questions about <a href="/calc1-001/lecture/297"><i class="icon-film"></i>&nbsp;Riemann sums</a>,</li>
<li>two questions about <a href="/calc1-001/lecture/291"><i class="icon-film"></i>&nbsp;linearity properties</a> and <a href="/calc1-001/lecture/287"><i class="icon-film"></i>&nbsp;the sign of the integral</a>.</li>
</ul>
<p>If you find yourself confused about the midterm or the exercises,
make sure to <a href="/calc1-001/forum/index">post a message on the
forum</a> so we can work together on it, or <a
href="/calc1-001/wiki/view?page=OfficeHours">talk it through with us
during office hours</a>.  <strong>Hang in there!</strong></p>