diff --git a/College/AU 26/MATH 224 (Multivar)/Chapter 15.3 - Double Integrals in Polar Coordinates.md b/College/AU 26/MATH 224 (Multivar)/Chapter 15.3 - Double Integrals in Polar Coordinates.md index d556e4a..cc9b682 100644 --- a/College/AU 26/MATH 224 (Multivar)/Chapter 15.3 - Double Integrals in Polar Coordinates.md +++ b/College/AU 26/MATH 224 (Multivar)/Chapter 15.3 - Double Integrals in Polar Coordinates.md @@ -5,5 +5,15 @@ $$\begin{gather} a\leq r\leq b \\ \alpha \leq \theta \leq \beta \end{gather}$$ +![[PolarRegion.excalidraw]] -if a function is defined as $f(x,)$ \ No newline at end of file +if a function is defined as $f(x,y)$ and the region it is integrated over is polar the integral will often look similar to +$$\int_{\alpha}^\beta \int_{a}^b f(r\cos(\theta), r\sin(\theta))r \ dr d\theta$$ + + +Find the volume of the solid below $z=1-x^2-y^2$ and above the first quadrant on the xy plane + +rearranging the equation gives $x^2+y^2=1-z$ which shows that each horizontal slice of the function is a circle centered at $(0,0)$ with a radius of $\sqrt{ 1-z }$ + +since the slice of the equation at $z=0$ forms the circle $x^2+y^2=1$ the bounds +$$$$ \ No newline at end of file diff --git a/Excalidraw/Drawing 2026-10-07 11.39.00.excalidraw.md b/Excalidraw/Drawing 2026-10-07 11.39.00.excalidraw.md deleted file mode 100644 index 3d8a720..0000000 --- a/Excalidraw/Drawing 2026-10-07 11.39.00.excalidraw.md +++ /dev/null @@ -1,197 +0,0 @@ ---- - -excalidraw-plugin: parsed -tags: [excalidraw] - ---- -==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving' - - -# Excalidraw Data - -## Text Elements -## Embedded Files -eb5f6a288d26a7a56e5db30c91f20873a1fdd6d6: $$\color{blue} \theta = \alpha$$ - -f29a5039d81fd6c72a5f96fce01c57e3dc17dd50: $$\color{orange} \theta = \beta$$ - -%% -## Drawing -```compressed-json -N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATLZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHALRY8RMpWdx8Q1TdIEfARcZgRmBShcZQUebR4ADm0ARho6IIR9BA4oZm4AbXAwUDBSiBJuCABrABZMAGV6tiMAfXoATh52gCsAUXx6ADEAJQBNKqR+MthESqIOSaLITG5n - -ZOSkgGYExOSeAFYpyBhVzc39+IB2dpv1mpuABh5NmqOIChJ1bge3yQRCZTSbj7GqbbTtBI7BJ7Q5LCDWZTBb5vZhQUhsCYAYTY+DYpEqAGJkghicS0qVIJpcNgqsp0UIOMRsbj8RI0dZmHBcIEcuSygAzQj4fD1WBIiSCDx8gRojEIADqn0k3D4cNR6Imopg4vQkoqb3pgIWzDyaGSbzYXOwahOZoePzhdOEcAAksRTah8gBdN788hZN3cDhCYVv - -QiMrCVXAPaUQemM40ekoU6DwcS8JYAXxRCAQxG4yXO1weADZ2g9YSnGCx2FwzSWS29q6xOAA5ThibigniXS6bdq9sPMAAiGSgee4/IIYTemmEjP6mWyuSDIfwbyEcGIuHH+bN10LEJeJZ4NVVKewgR3E/37UPCRe/ZqDpT8yqq9DcNxNJvqCn+BnOE4DYcMVzQQoKTACDIJfGCljAH14OgilYIpEEwQhKEYXgr0lkQlNUW5KAACFw0cDhlA/dc4W - 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-W4vKRL9NwUp0EMI9XRB1CtctTUvUKnXkRQgLiHXKFypEvwBwyyMscBmAPMuxAcgejU49sgIQAm9KQK1Aoh0PPLR5AyEC1DzKTkcBGYBOPEgC8ErSscrgBGIHso5LKykAxa8GgU8trAJylyAX4ogcMBELwwOABrLRwMwCArcAGsrYBzdCMv4B8dHjSTBMwEAEzAgAA=== -``` -%% \ No newline at end of file diff --git a/Excalidraw/PolarRegion.excalidraw.md b/Excalidraw/PolarRegion.excalidraw.md new file mode 100644 index 0000000..1a88ec8 --- /dev/null +++ b/Excalidraw/PolarRegion.excalidraw.md @@ -0,0 +1,659 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving' + + +# Excalidraw Data + +## Text Elements +## Embedded Files +eb5f6a288d26a7a56e5db30c91f20873a1fdd6d6: $$\color{blue} \theta = \alpha$$ + +f29a5039d81fd6c72a5f96fce01c57e3dc17dd50: $$\color{orange} \theta = \beta$$ + +58cec4a2026c7c65a7d7bc279dff6d9209e9e8e9: $$r=a$$ + +9f38209425207e9acdbdb35901a64fe5e123a19d: $$r=b$$ + +%% +## Drawing +```compressed-json +N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATLZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHALRY8RMpWdx8Q1TdIEfARcZgRmBShcZQUebR4ADm0ARho6IIR9BA4oZm4AbXAwUDBSiBJuCABrABZMAGV6tiMAfXoATh52gCsAUXx6ADEAJQBNKqR+MthESqIOSaLITG5n + +ZOSkgGYExOSeAFYpyBhVzc39+IB2dpv1mpuABh5NmqOIChJ1bmTtdoA2RJ/BI1YH7AF/EFvSQIQjKaTcTZ/C48S4A9qXFE1fYJS77TZvazKYLcB5vZhQUhsCYAYTY+DYpEqAGJkghWay0qVIJpcNgqspKUIOMRafTGRIKdZmHBcIEcpyygAzQj4fD1WDEiSCDwKgQUqkIADqn0k3D4Swg5MpE3VME16G1FTegrhC2YeTQyTebBl2DUJ09D1JFoFw + +jgAEliB7UPkALpvRXkLKR7gcISqt6EYVYSq4B66iCC4Vu6MlLnQeDiXhLAC+ZIQCGI33O1wef3aD0OFsYLHYXDQKISbx7rE4ADlOGJvu19ri/psHujM8wACIZKCN7iKghhN6aYTC/qZbK5Arxi1CODEXAbpue67JTbtHHPjFei3YQI3zf39qP58vE+NTBuW8xVKm6b4G89J8j+qDbvgu4WnAbBZqeaCFFyYCYVhIG4UsYDnlhOFcmsvwAgkQIgli + +4KQgRYCIsiqJdBilxYjieIEbGSxEWU5KylAABCWaOBwygQRmFrZMQwnClm4loGmknlvxpBQAAgqQlIUNCuB3qgSlQVJwqadpun6YZbwofgMDKJwW47ggUKhAAKlgUAADJZuBaAIWERR1kUZaQBUEhGM4lwABqkGOhoAFq9EYcCYJokXKFgzjqZcowFjMVYQPMizlisaDODwyR/L8C44skGILp0+yvBaAaoBFFz7GCnSbJcCQPCCBzmuWHzEF8aCb + +DwNTaH81zoskNFBk8fxQjCcJQGavx9XcQLTfsna1UtFqEvaeF6taCCigyzLsmyRVlDyfKhkKIp0pdErkBw0qyieBbKqqtr2padJOhaVoGsaI2mgOZL6jaGr5Y6TbOsIrrhNG77lj6vL+t8QZvI9EZRmeCZJggKaKZBmbZiV6C4KkSNPSW3DBZAeVmrW9Zwe0NTjR2NXDkwo79qg2xduWI59hOHBTp6PB/A8uK4sua7BLeDmIU5Fr7k9R5ZDk0Zxm + +8l7Xqrv7/gkvU1I+lxvJ+IQm6gtV/k+CTnEGlwnQV3kSUZoFsLB+l+Rr5YoWh+sESRpQnZHXFHNh9HjZN003Jcc1ggtsux2AZUbZbc3bbie2olxPHQwJsmiQpBkU8ZMkifJ3ul2pplsDpIQWdX5bSc3rd6Q3yF0rZ9m+Y5znMG5mCeV7w/qwFUzBeU+kQAAqgAjgA4r0gz6AAEgMegec4DwtO0zj1M4ABq1K5ZWcxZrdyyrOslzkdR3Wi2ULVtSk + +T6In8c2PM8JqQ0TTcD/L8VEYIDjLVhPCAciRtA1C6ECRWh0xLHWhmdC64p0AshuhyPcvJ+RFmemKSokoPoyjlGtBMKo1Rw0qAjAsoMJjg1GrwDBBp/rwyBojC0LpJCM09N6X02NAwe3xpGfWvFICJlwMmduykygiRzBIXAPACzEMEagZmFZZgDnZiDBs+lki7TxEGTq/NexDwdvcSxgtJbS1QKiF2ct2zWwtIQVc644KBz3AeYgOsTxSMNleb8xi + +HzO3OL1CqNsvz20dmbc4eJubQSnlXRRkAYITADiPfuocCjh3olHMADwY6FKwmAMB6IkQAnfqRHg8DEH/A4sXLk0jLRRDUuXeu5MMkYGFN0sSfcVKdI0lpFu5lhllC7uMnuCifZlGsoPIWviLSSFcu5LyHAfLwUcrPIKHjF6DEVOpQYkVBg1AAIrhhqESfAmAKBGBcgAaU0PgZw189HoEKgWamaxyrxDmsCN+bxP6XGfrVLEDSnwAJeG8YabDypxE + 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