Calculus

Take-home Exam test

MA102

Dr. Ali Yousef

Instruction: Please solve the following problems showing the logical steps.

Each part worth 5 points.

Q1. a. Find the area of the region that lies inside 𝑟 = 1 + 𝑐𝑜𝑠(𝜃) and outside 𝑟 = 2 − 𝑐𝑜𝑠(𝜃).

b. Determine the arc-length of the function 𝑟 = 1 + 𝑐𝑜𝑠(𝜃) from 0 ≤ 𝜃 ≤ 𝜋

3 .

c. Find the slope of the tangent line of the graph 𝑟 = 1 + 𝑐𝑜𝑠(𝜃) at 𝜃 = 𝜋

4 .

d. Find the equation of the tangent line of 𝑟 = 1 + 𝑐𝑜𝑠(𝜃) at 𝜃 = 𝜋

4 .

Q2. Consider the parametric equations given by:

C: 𝑥 = 3 − 𝑐𝑜𝑠3(𝑡) 𝑦 = 4 + 𝑠𝑖𝑛(𝑡) , 𝑡 ∈ [0, 𝜋]

a. Eliminate the parameter 𝑡, and find a relation between 𝑥 𝑎𝑛𝑑 𝑦.

b. Find the arc length of the parametric curve.

c. Determine the area under the curve given by the parametric equations.

Q3. Determine the surface area of the solid obtained by rotating the parametric curve about the 𝑥 −

𝑎𝑥𝑖𝑠. C: 𝑥 = 𝑐𝑜𝑠3(𝑡) 𝑦 = 𝑠𝑖𝑛(𝑡) , 𝑡 ∈ [0, 𝜋

2 ].

Q4. Determine the equation of the tangent line to the parametric curve at the point (2, 0).

C: 𝑥 = 𝑡2 + 1 𝑦 = 𝑡3 − 𝑡

Good Luck