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Find the volume of the solid formed by revolving the region bounded by y=(x-2)² and y=x about the y -axis.
You are looking at producing a 3D model of a solid shape you want to manufacture. The shape is a cone which is bounded by the line 𝑦=1/2𝑥 rotated about the 𝑥−𝑎𝑥𝑖𝑠 between 𝑥=0 𝑎𝑛𝑑 𝑥=5?
Use calculus to find its volume.
1. The energy 𝑖, of an inductor with inductance 𝐿 is given by
𝑖= 12𝐿∫𝑡2𝑒−𝑡10𝑑𝑡
For 𝐿=(1 ×10−3)𝐻, Find 𝑖.
Locate and classify the stationary points of the following:
(i) f(x,y) = 4xy + x4 - y4
(ii) f(x,y)= xy + 2/x + 4/y, x>0, y>0
if f is continuous for all x and, ∫ f(x)dx and ∫b/a f(x) dx are equal which one of the following statement is correct, 1. the function f is the constant e ,2. the function f is not differentiable, 3. the function f is the constant 0 , 4. the function f is any arbitrary constant c ,5. the function f is the identity function
Find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis and Sketch the graph.
Circular Disk Method:
- R:y=√𝑥, 𝑦=4, 𝑦−𝑎𝑥𝑖𝑠 axis of revolution: 𝑦=4
- 𝑅:𝑦=𝑥^2, 𝑥=3, 𝑥−𝑎𝑥𝑖𝑠 axis of revolution: 𝑥=3
Circular Ring Method:
- 𝑅: 𝑦=𝑥^2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
- 𝑅:𝑦^2=4𝑥, 𝑥^2=4𝑦 axis of revolution: 𝑥=4
Cylindrical Shell Method:
- 𝑅:𝑦=𝑥^2, 𝑦=2𝑥 axis of revolution: 𝑥−𝑎𝑥𝑖𝑠
- 𝑅: first-quadrant region bounded by 𝑦=4𝑥 and 𝑥^3=𝑦 axis of revolution: 𝑥=2
- An oil tank in the shape of a sphere has a diameter of 60 ft. How much oil does the tank contain if the depth of the oil is 25ft?
Find the centroid of the solid generated if the region bounded by y = 2- x^2, x=0, and y=0 is revolved about the y-axis.
lim (3x^2+4/ 5x^4+ 7x^2+1)
x→∞
Find the mass of the object, which is in the form of a sphere of radius √5cm, centred at the origin. The density at any point is given to be the constant 2.
Show that the value of the integral
∫ [(x^2+3y)dx + (5x-3y^2)dy]
C
where C is the ellipse x^2/a^2+y^2/b^2=1, is twice the area enclosed by C.
