Determine the arc length of a rose curve

Reference no: EM132296227

Mathematics Questions -

Q1. (a) Derive a formula for the surface area of an object that is created by rotating a function f(x) around:

1. the x-axis with y ≥ 0

2. the y-axis with x ≥ 0

You will need to clearly show how you have chosen to break the surface up into tiny pieces and what high school geometry is needed to find the area of these tiny pieces.

(b) Confirm that your formula provides the expected surface areas of cylinders, cones and spheres by considering an appropriate function f(x).

(c) Show that a slice of a sphere of width Δx will have the same surface area regardless of where that slice is taken, parallel to the y-axis. To put it another way, a spherical loaf of bread will have an equal amount of crust on all slices - yes, even the ends.

Q2. A rose by any other name...

(a) The equation for arc length we have seen in lectures is:

_a∫^b√(1 + (dy/dx)²) dx

Convert this to an arc length of a curve that is given in polar coordinates.

(b) Investigate 'rose curves' and summarise what they are.

(c) Determine the integral that should be used to determine the arc length of a rose curve and explain why a solution will not be possible.

(d) Determine the integral that should be used to determine the area of a rose petal and evaluate the area of the curve produced for your favourite rose curve.

Q3. (a) What is a logistic sigmoidal function and what can it be used for? Assign values to the necessary parameters in the logistic sigmoidal curve and use these to complete the following questions:

(b) State the domain of the standard logistic sigmoidal function.

(d) State the inverse of the standard logistic sigmoidal function?

(e) State the domain and range of the standard logistic sigmoidal function f(x) its inverse f^-1(x)?

(f) Plot f(x) and f^-1(x) on the same axis.

Q4. In class we had a question regarding the spherical coordinate system:

Given that

x = r cosθ sinφ

y = r sinθ sinφ

z = r cosφ

with 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π

"Why don't we have 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π instead?"

(a) Explain why this would not work.

(b) If you really wanted the bounds suggested how could you make it work?

Q5. How well do you really understand integral calculus?

Consider the above circles, that have each been divided into infinitely small/infinitely many parts (use your imagination). This is an investigative question and marks will be awarded for depth of solution.

(a) Show that in each case the area of the circle, with radius r, calculated with an integral is what you expect.

(b) How could you divide a square? Hence, show that the area of a square of side length X is what you expect.

Q6. Complete one of the past exams available on the unit web page. Note that not all questions will be possible prior to the due date as we have not yet covered all the material. If you are unsure of which questions are required come and see me.

Reference no: EM132296227

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