Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1....

Question

Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

Accepted Answer

Determine the area enclosed by the ellipse, X2 divided by 16 plus Y2 divided by 25 equals 1. We have 4 possible answers, being 40 pi, 2016 pi, or 8 pi, all in square units. Now, this ellipse is centered at the origin, which means we can divide our ellipse into 4 symmetrical and equal sections. So we can say our total area A will be 4 multiplied by the area. Of the first quadrant. Since every area will be the same. So now to solve this, let's first find a couple of things. Let's first find. Our ex intercepts. X2 divided by 16 + Y2 divided by 25 equals 1. For this, then, we know y will be equal to 0. For the X intercepts. This means we have x2 divided by 16 equals 1, or x2 equals 16, x equals plus or minus 4 when we take the square root. Meaning our vertices for X are 40 and -40. This helps us find the bounds of our integral. Now we need the equation for our integral. We can find this by solving for y. We have X2 divided by 16 plus Y2 divided by 25 equals 1. We can subtract X2 divided by 16 to get Y2 divided by 25 equals 1 minus X2 divided by 16. Now we can multiply everything by 25 and factor out 16. We get y squared equals 25 divided by 16 multiplied. By 16 minus x 2 We can then take the square root of everything. Why then Becomes plus or minus 5/4 square root, 16 minus x 2. So we can now set up our integral. Because we're just using quadrant 1, we can say A equals 4 multiplied by the integral. From 0 to 4 for our X intercepts. Of The square root 16 minus X2 DX. Now this is our area integral. Let's solve this. We know the standard integral for the square root of A2 minus X2 DX is equivalent to the following. X divided by 2 square root a squared. Minus x 2 Plus A 2 divided by 2 sin inverse. Of X divided by a plus C. Now in our case, A equals 4. So, We can then write out our integral. We have the integral 16 minus x 2 under the square root. S equals To x divided by 2. Square root 16 minus x 2. Plus 4 2 divided by 2, which is just 8. Sign inverse x divided by 4 plus C. Now from here. We can simplify even further. Keep in mind from before. We did have a 5/4 in our equation. So we end up getting 5 of us are constant outs. Multiplied by X divided by 2, 16 minus X2, plus 8 inverse, X divided by 4. This is all From 0 to 4. We can thus simplify this by plugging in our values. When we do that. We get 5 multiplied by 8 sin inverse. Of 1 minus 0 Which just gives us 5 multiplied by 8, multiplied by pi. Divided by 2 or 20 pi. This will be in square units. This is the answer for our area, which means if we look at our answers, this is answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

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