Calculus Answers

(a)Find the area of the region enclosed by the parabola 𝑦 = 2𝑥 − 𝑥

2 and the 𝑥 axis.

(b) Find the value of 𝑚 so that the line 𝑦 = 𝑚𝑥 divides the region in part (a) into two

regions of equal area.

Verify that the function 𝑦 = 𝑐1𝑒 (−𝑘+2𝑖)𝑥 + 𝑐2𝑒 (−𝑘−2𝑖)𝑥 is a solution to 𝑦 ′′ + 2𝑘𝑦 ′ + (𝑘 2 + 4)𝑦 = 0.

Evaluate the line integral

∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 ,

where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦^2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1]. 

Question 4 Evaluate the line integral:

(i) of 𝑇(𝑥) = 4𝑥^3 along the line segment from (−2,1) to (1,2).

(ii) where the curve 𝐶 is parameterized through 𝑥(𝑡) = cos 𝑡, 𝑦(𝑡) = sin 𝑡 and 𝑧 = 𝑡^2 with 𝑡 ∈ [0,2𝜋] of ∫(𝒚𝒅𝒙 + 𝒙𝒅𝒚 + 𝒛𝒅𝒛) 𝐶

(iii) ∫ 𝐅(𝑥, 𝑦, 𝑧) ⋅ ⅆ𝐫 𝐶 , where 𝐹(𝑥, 𝑦, 𝑧) = (5𝑧^2 , 2𝑥, 𝑥 + 2𝑦) and the curve 𝐶 is given by 𝑥 = 𝑡, 𝑦 = 𝑡^2, and 𝑧 = 𝑡^2 with 𝑡 ∈ [0,1]

Question 3

Determine the length of the curve 𝑥 = 𝑦^2 /2 for 0 ≤ 𝑥 ≤ 1/2 . Assume 𝑦 positive.

Question 2

(i) Find the volume integral of the scalar field 𝜙(𝑥, 𝑦, 𝑧) = 𝑥^2 + 𝑦^2 + 𝑧^2 over the region 𝑉 specified by 𝑥 ∈ [0,1], 𝑦 ∈ [1,2], and 𝑧 ∈ [0,3].

(ii) Find the gradient of the scalar field 𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧 and evaluate it at the point (1,2,3). Hence, find the directional derivative of 𝑓 at this point in the direction of the vector (1,1,0).

Question 1

(i) Construct the contour plot of the scalar field 𝑇: ℝ2 → ℝ such that 𝑇(𝑥, 𝑦) = 𝑥 2 − 𝑦.

(ii) Sketch the vector field 𝑉 ∶ ℝ2 → ℝ2 defined as follows 𝑉(𝑥, 𝑦) = (𝑥 + 𝑦, −𝑥). 

Convergence test for "\\displaystyle\\sum_{n=1}^\\infty \\frac{sin(n)}{n}".

Given the domain(-2,-1,0,1,2), determine the range for each expression.Use a table of values

Use the definition of the derivative to differentiate V=(4)/(2)\pi r^(3).