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# (a) $$2 \pi\left[b^{2}+\frac{a^{2} b \sin ^{-1} \frac{\sqrt{a^{2}-b^{2}}}{a}}{\sqrt{a^{2}-b^{2}}}\right]$$(b) $$2 \pi a^{2}+\frac{2 \pi a b^{2}}{\sqrt{a^{2}-b^{2}}} \ln \frac{\sqrt{a^{2}-b^{2}}+a}{b}$$

## a) $$2 \pi\left[\frac{a^{2} b}{\sqrt{a^{2}-b^{2}}} \arcsin \left(\frac{\sqrt{a^{2}-b^{2}}}{a}\right)+b^{2}\right]$$b) $$2 \pi a^{2}+\frac{2 \pi a b^{2}}{\sqrt{a^{2}-b^{2}}} \ln \left(\frac{a+\sqrt{a^{2}-b^{2}}}{b}\right)$$

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Applications of Integration

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pants, Clara. So when he married here. So we're going to use the parametric equations, which is X is equal to a co sign of data. And why is equal to be sign of data Data's between zero and Pi. We get D s to be equal to the square root of D X over de Vita Square Plus, do you? Why Over a deed Zeta square, which is equal to the square root of negative A sign Data square plus be co sign, you know, square, but just equal to the square root of a square sign square plus B square Coastline square. We're gonna put this into RS equation. So to a pie from zero to high, he signed data. We're gonna place this right here. This is equal to two pi b I'm zero to pie. Sign of data square root of a square minus square. CO signed square. It must be a square. Resigned square. Be fatal. This becomes equal to two pi B from zero to pine sign data square of the square plus the square minus a square. Who's saying square Seita we get a you value of co sign. So do you have to be cool to negative sign Zeta so as his equal to negative to Pai Bi I'm one negative one square root square cost Be square minus a square You square do you which is equal to or pie be from 0 to 1 square root of a square plus be square minus a square You swear to you we know that a square minus the square is equal to a square times east where so this is equal to four pi b from 0 to 1 square root of in square minus key square e square you square do you which is equal to four pi maybe being from 0 to 1 square root of one divided by een square minus new square Do you Using the table of integral sze we get for a pie You square you over two square root of one divided by e square minus years square plus one divided by the square over too inverse sine of you over one divided by t from 0 to 1, just equal to four pi a b e square root yeah of one minus square over to e plus one over to e square, inverse sine of E this becomes equal to two pi be be over a plus won over e in birth Sign of me her part being We're doing a similar approach but we're going to use the minor absence So we get data to be between negative pie house and positive pie hands. So you used to seem Parametric inflation's, which is access equal to a co sign and want, is equal to be signed. So yet the square root of a square signs where plus squared coastline square. But for us, we're going to use to pine into girl from negative pie house to positive Pi House a co sign data square root of a square Sign square has B square co sign square de fita which becomes equal Thio too fine, then to grow from negative pie house to positive pie halves A co signed data squarely of a square minus B square Sign square must be squared Defeat We're using you to be equal to sign of data. So do you Is equal to co sign data data so us becomes equal to two Pipe, eh from negative 1 to 1 square root of a square minus b square you're square plus B square. See you, which becomes equal to for pie, eh? Square of a square minus b square from 0 to 1 square root of you Square plus B square over in square minus B squared. Do you You know, not the squarely of a square minus B square is equal to 80. So this becomes equal to four high a square e from 0 to 1 square root Uh, you square waas beat about it by anything square d d you. We're using the table of integral sze and we get this to be equal to four Fine e swear e times you over two square root of B divided by a E square We'll ask you square less be divided by a e square over too well and of you plus square root of you square plus be divided by 80 square from zero 21 all areas some of here to make more room continuing on This becomes equal to for pie in square e times one how square is a P divided e square plus one plus so plus one waas be divided by a C flare over to Owen have one plus square root of one plus Be divided by a E square minus for pie. Be square turns be divided by eight square over too. Oh, and be divided by a E. This becomes each of two to pie finds a flare Must be square over. How on, uh, he plus one over. Be divided by becomes equal to two pine terms a square past. Be square over E Oh, um of one plus over the square root of farm miners E square.

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