63–74. Arc length of polar curves Find the length of the follo...

Question

63–74. Arc length of polar curves Find the length of the following polar curves.

{Use of Tech} The complete limaçon r=4−2cosθ

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Compute the area of the part of the unit disk x 2 plus Y 2 is less than or equal to 1 that lies to the left of the vertical line. X is equal to -12 using polar coordinates. OK. So it appears for this particular prompt we're asked to determine what the area of part of the unit disk is using the following equation that is given to us X 2 plus y to the power 2 is less than or equal to 1 that lies to the left of the vertical line, X is equal to -12, and we're asked to do this using polar coordinates. So now that we know that we're ultimately trying to determine what the area is, let us read off our multiple choice answers to see what our final answer may be. A is pi divided by 3 minus the square root of 3 divided by 4. B is pi divided by 2 minus the square root of 3 divided by 2. C is 2 pi divided by 3 minus the square root of 3 divided by 2. And D is pi divided by 3 minus the square root of 3 divided by 2. OK. So our first step that we need to take is we need to recall how to find the central angle, and as we should recall based off of our previous knowledge, the region is the region that we're focusing our attention on is a circular segment of the unit circle that has been cut off by the vertical cord. X is equal to -12, and the distance from the center to the cord will be D is equal to 12, and the radius, which will denote as R for radius is equal to 1. So that will mean, based on our prior knowledge, the central angle is subtended by the chord will be written as theta. The angle theta is equal to 2 multiplied by inverse cosine of D divided by capital R, which is equal to 2 multiplied by inverse cosine of 12, which will be equal to 2 multiplied by pi divided by 3, which is equal to 2 divided by 3. Fantastic. Our second step now is we need to recall how to set the area formula. We need to recall how we're going to set up the area formula. And as we should recall, based on our prior knowledge, the area of the circular segment with the central angle theta, which will be in units of radiance. And the radius R will equal the sector area minus the isosceles triangle area. So that means we could write that the area A is equal to 1/2 multiplied by R2 multiplied by the minus 1/2 multiplied by R2 multiplied by sin of theta. Which will be equal to R2 divided by 2 multiplied by the parenthesis of theta minus sine of theta. Fantastic. So now our third and final step is we need to evaluate the area. So we need to recall a note once again that R is equal to 1, and that theta is equal to 2 pi divided by 3. And we need to recall a note that sine of theta is equal to sign of drum roll, 2 pi divided by 3, which will be equal to the square root of 3 divided by 2 when we perform the evaluation. So that means that the area A will be equal to, when we plug in all of our known variables, A will be equal to the area A is equal to 1/2 multiplied by the parenthesis of 2 divided by 3 minus the square root of 3 divided by 2, which will be equal to pi divided by 3. Minus the square root of 3. Divided by 4, and that's it. We've officially solved for this problem. Hooray, we did it. So looking at our multiple choice answers, our correct and final answer without a doubt will be multiple choice answer letter A, which once again is pi divided by 3 minus the square root of 3 divided by 4. And that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

63–74. Arc length of polar curves Find the length of the following polar curves.{Use of Tech} The complete limaçon r=4−2cosθ

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63–74. Arc length of polar curves Find the length of the following polar curves.

{Use of Tech} The complete limaçon r=4−2cosθ