Tyler Clarke
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<p class="paragraph-P1">Mathin’ a Cos Rose</p>
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<p class="paragraph-P2">by Tyler Clarke</p>
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<p class="paragraph-P2"> </p>
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<p class="paragraph-P2">Got me a multivariable calculus exam coming up soon. Most of the material is pretty simple – lagrange multipliers, for instance (I’ll be writing something about those soon) – but I’ve got one particular problem regarding integrating with radial coordinates that’s hard and interesting.</p>
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<div class="paragraph-P2">I’m asked to find the area of a single loop of a <span class="text-T1">rose</span><span class="text-T2"> described in radial coordinates by </span>
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</math> </span><span class="text-T2">. </span><span class="text-T3">This is difficult to visualize, but luck hath smiled upon us, and Desmos supports radial coordinates:</span></div>
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"/></div><div style="clear:both; line-height:0; width:0; height:0; margin:0; padding:0;"> </div><span class="text-T3"/></div>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||
|
||
<p class="paragraph-P3"><span class="text-T2">Not too bad. The maximum extent would appear to be 5, which makes sense, because cosine maxes out at 1 and we multiply by 5. I was a bit curious to see if the multiplier of theta controls the number of loops – as it turns out, the relation is a bit complicated, and I won’t get into that now.</span></p>
|
||
|
||
<!--Next 'div' was a 'text:p'.-->
|
||
<div class="paragraph-P4"><span class="text-T4">It’s looking to me like the best solution here is to integrate for </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mo stretchy="false">≤</mo>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">≤</mo>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</mrow>
|
||
</math> </span><span class="text-T4">, to grab that petal in the first and fourth quadrants, but I’m worried that that isn’t a </span><span class="text-T5">general</span><span class="text-T4"> enough solution: other roses that I’ve got to graph might not be so conveniently situated. The more general way I can think of is integrating for all theta and dividing by the number of petals – which isn’t necessarily easily predictable for all theta, but is a damn sight better than just </span><span class="text-T5">hoping</span><span class="text-T4"> my hand-drawn sketches appropriately capture the useful quadrants. </span><span class="text-T6">Regardless, let’s do the first idea first, and see how it works out.</span><span class="text-T4"> In this case, we </span><span class="text-T5">know</span><span class="text-T4"> there are 3 petals, so the integral should be </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object3"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</munderover>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mn>0</mn>
|
||
<mrow>
|
||
<mn>5</mn>
|
||
<mi>cos</mi>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</munderover>
|
||
<mi>r</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mi mathvariant="italic">dr</mi>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</math> </span><span class="text-T4">. Why </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object4"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mi>r</mi>
|
||
<mi mathvariant="italic">dr</mi>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</math> </span><span class="text-T4"> instead of just </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mn>1</mn>
|
||
<mi mathvariant="italic">dr</mi>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</math> </span><span class="text-T4">? Because of the shape of the mesh we’re integrating; I won’t elaborate, but only because </span><a href="https://tutorial.math.lamar.edu/classes/calciii/dipolarcoords.aspx" class="text-Internet_20_link"><span class="text-T4">Paul's Online Math Notes</span></a> <span class="text-T7">already did.</span></div>
|
||
|
||
<!--Next 'div' was a 'text:p'.-->
|
||
<div class="paragraph-P5">Integrating this isn’t too hard. The inner integral is simply
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mrow>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mn>0</mn>
|
||
<mrow>
|
||
<mn>5</mn>
|
||
<mi>cos</mi>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</munderover>
|
||
<mi>r</mi>
|
||
</mrow>
|
||
<mi mathvariant="italic">dr</mi>
|
||
<mo stretchy="false">→</mo>
|
||
</mrow>
|
||
<mrow>
|
||
<msubsup>
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<msup>
|
||
<mi>r</mi>
|
||
<mn>2</mn>
|
||
</msup>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||
</mrow>
|
||
<mn>0</mn>
|
||
<mrow>
|
||
<mn>5</mn>
|
||
<mi>cos</mi>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</msubsup>
|
||
<mo stretchy="false">→</mo>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<msup>
|
||
<mi>cos</mi>
|
||
<mn>2</mn>
|
||
</msup>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span>, giving us
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object7"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</munderover>
|
||
<msup>
|
||
<mi>cos</mi>
|
||
<mn>2</mn>
|
||
</msup>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</math> </span>. <span class="text-T8">This is kind of a nasty trig integral, one which I very much don’t want to derive myself, but that’s what Wolfram is for! Wolfram kindly tells me that my disgusting integral is </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<msubsup>
|
||
<mrow>
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>24</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mrow>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">+</mo>
|
||
<mi>sin</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||
</mrow>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</msubsup>
|
||
</math> </span><span class="text-T8">. Because </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object9"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mi>sin</mi>
|
||
<mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
<mo stretchy="false">=</mo>
|
||
<mn>0</mn>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span><span class="text-T8">, this works out nicely to… </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>24</mn>
|
||
</mfrac>
|
||
<mn>6</mn>
|
||
<mrow>
|
||
<mi>π</mi>
|
||
<mo stretchy="false">=</mo>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>4</mn>
|
||
</mfrac>
|
||
</mrow>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
</math> </span><span class="text-T8">. </span><span class="text-T9">This is wrong.</span></div>
|
||
|
||
<p class="paragraph-P6"><span class="text-T10">I’ve kept the wrong work up because it’s </span><span class="text-T11">subtly</span><span class="text-T12"> wrong, and (I think) does a pretty good job of illustrating one of the most annoying problems with polar coordinates. Do you see it?</span></p>
|
||
|
||
<!--Next 'div' was a 'text:p'.-->
|
||
<div class="paragraph-P7"><span class="text-T13">The problem lies in how cosine behaves with multipliers. </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mi>cos</mi>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span><span class="text-T13"> makes a full trip from 1 to 0 in </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object12"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mn>0</mn>
|
||
<mo stretchy="false">≤</mo>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">≤</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
</math> </span><span class="text-T13">; </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object13"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mi>cos</mi>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span><span class="text-T15"> makes </span><span class="text-T16">three</span><span class="text-T15"> such trips in the same span. And our rose has three peaks.</span></div>
|
||
|
||
<!--Next 'div' was a 'text:p'.-->
|
||
<div class="paragraph-P6"><span class="text-T17">By some horrible mathemagic, integrating what should be a half-turn integrates </span><span class="text-T18">the whole pattern</span><span class="text-T17">. As it turns out, this is true for every cosine rose: regardless of the multiplier, a full span of pi (not 2pi!) covers the entire shape. Because we only want one petal, we have to cut our</span><span class="text-T18"> bounds</span><span class="text-T19"> </span><span class="text-T17">in third: </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object14"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>6</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>6</mn>
|
||
</mfrac>
|
||
</munderover>
|
||
<msup>
|
||
<mi>cos</mi>
|
||
<mn>2</mn>
|
||
</msup>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">→</mo>
|
||
</mrow>
|
||
<mrow>
|
||
<msubsup>
|
||
<mrow>
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>24</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mrow>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">+</mo>
|
||
<mi>sin</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||
</mrow>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>6</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>6</mn>
|
||
</mfrac>
|
||
</msubsup>
|
||
<mo stretchy="false">→</mo>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>12</mn>
|
||
</mfrac>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span><span class="text-T20">. </span><span class="text-T21">And that’s right!</span></div>
|
||
|
||
<!--Next 'div' was a 'text:p'.-->
|
||
<div class="paragraph-P8"><span class="text-T17">What if we integrated the entire shape and divided by three, as mentioned above? This gives us actually the same integral, with a different multiplier and different bounds: </span>
|
||
<!--Next 'span' is a draw:frame. -->
|
||
<span id="Object15"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>1</mn>
|
||
<mn>3</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<munderover>
|
||
<mo stretchy="false">∫</mo>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</munderover>
|
||
<msup>
|
||
<mi>cos</mi>
|
||
<mn>2</mn>
|
||
</msup>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>3</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
<mi>d</mi>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">→</mo>
|
||
</mrow>
|
||
<mrow>
|
||
<msubsup>
|
||
<mrow>
|
||
<mrow>
|
||
<mrow>
|
||
<mfrac>
|
||
<mn>1</mn>
|
||
<mn>3</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>24</mn>
|
||
</mfrac>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mrow>
|
||
<mi>θ</mi>
|
||
<mo stretchy="false">+</mo>
|
||
<mi>sin</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||
<mrow>
|
||
<mrow>
|
||
<mn>6</mn>
|
||
<mi>θ</mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||
</mrow>
|
||
<mfrac>
|
||
<mrow>
|
||
<mo stretchy="false">−</mo>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
<mfrac>
|
||
<mi>π</mi>
|
||
<mn>2</mn>
|
||
</mfrac>
|
||
</msubsup>
|
||
<mo stretchy="false">→</mo>
|
||
<mfrac>
|
||
<mn>25</mn>
|
||
<mn>12</mn>
|
||
</mfrac>
|
||
<mi>π</mi>
|
||
</mrow>
|
||
</mrow>
|
||
</math> </span><span class="text-T17">. Exactly the same! Math is weird like that.</span></div>
|
||
|
||
<p class="paragraph-P8"><span class="text-T17">To avoid boundary hell in the future, I’m just going to integrate whole shapes and divide by petals. It’s much easier to wrap my head around why that actually works.</span></p>
|
||
|
||
<p class="paragraph-P8"><span class="text-T17"/></p>
|
||
|
||
<p class="paragraph-P8"><span class="text-T17">Stay tuned for Lagrange multipliers!</span></p>
|
||
</body>
|
||
|
||
</html>
|