Next we know that if we plug 2 π into our formula

Info icon This preview shows pages 7–9. Sign up to view the full content.

Next, we know that if we plug 2 π into our formula for the radius, the radius is once again 3. This means that the total angles that trace out this figure are from 0 to 2 π . So, we can solve our equation to find the area. 3 + sin ( 5 θ ) ¿ 2 dθ≈ 19 π 2 29.8451 1 2 ¿ 0 2 π ¿ b. We know that the smallest value of our radius of the function given for the star will be the largest radius that the circle can be, so we can do this by looking at the graph of the function 3 + sin ( 5 θ ) and find that the minimum value of the graph is 2. We can also know from the graph that the first 5 values of theta at these points where the radius equals 2 are 3 π 10 , 7 π 10 , 11 π 10 , 3 π 2 , 19 π 10 .
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

c. We know the formula for the arc length of a polar figure is length = α β . We also know from part a that the values of theta that trace out the figure are form 0 to 2 π . Now, we need to find f ' ( θ ) which from derivative rules we can find out is 5cos ( 5 θ ) . So now we can plug these values into our function and get 0 2 π . d. To evaluate the area of one of these pieces, we can subtract two integrals. First, we need to find our values for alpha and beta by setting the two polar equations equal to each other. 3.5 = 3 + sin ( 5 θ ) , 0.5 = sin ( 5 θ ) ,sin 1 ( 0.5 )= 5 θ 0.52 / 5 = θ . θ = 0.105, π 6 , and we can also verify this by looking at a graph. Now, we can set up our integrals. 3.5 ¿ 2 d θ≈ 0.506 ¿ 1 2 ¿ 3 + sin ( 5 θ ) ¿ 2 0.105 π 6 ¿ 1 2 ¿ 0.105 π 6 ¿ . e. To do this, we first set up our derivative as it is shown in the problem, then we find what values of theta make that derivative equal 0 between 0 and 2 π . So,
Image of page 8
we do: y ' ( θ ) x' ( θ ) = 5sin ( θ ) cos ( 5 θ )+ cos ( θ )( sin ( 5 θ )+ 3 ) 5cos ( θ ) cos ( 5 θ )− sin ( θ )( sin ( 5 θ )+ 3 ) = 0 . Now, to solve for the values of theta for which this is true we can graph this and find how many times it crosses the x-axis between 0 and 2 π , which is 8.
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern