Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
1 answer:
Answer:
V = 34,13*π cubic units
Step-by-step explanation: See Annex
We find the common points of the two curves, solving the system of equations:
y² = 2*x x = 2*y ⇒ y = x/2
(x/2)² = 2*x
x²/4 = 2*x
x = 2*4 x = 8 and y = 8/2 y = 4
Then point P ( 8 ; 4 )
The other point Q is Q ( 0; 0)
From these two points, we get the integration limits for dy ( 0 , 4 )are the integration limits.
Now with the help of geogebra we have: In the annex segment ABCD is dy then
V = π *∫₀⁴ (R² - r² ) *dy = π *∫₀⁴ (2*y)² - (y²/2)² dy = π * ∫₀⁴ [(4y²) - y⁴/4 ] dy
V = π * [(4/3)y³ - (1/20)y⁵] |₀⁴
V = π * [ (4/3)*4³ - 0 - 1/20)*1024 + 0 )
V = π * [256/3 - 51,20]
V = 34,13*π cubic units
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The pairs of points are:
a. A(-6,-4), B(-3,-1)
b. C(3.5,1), D(-4,2.5)
c. X(5,-5), Y(-5,5)
To find:
The distance between the pair of points by using the distance formula.
Solution:
Distance formula:
a.
Using the distance formula, the distance between the points A(-6,-4) and B(-3,-1) is:
Therefore, the distance between the points A(-6,-4) and B(-3,-1) is about 4.2 units.
b.
Using the distance formula, the distance between the points C(3.5,1) and D(-4,2.5) is:
Therefore, the distance between the points C(3.5,1) and D(-4,2.5) is about 7.6 units.
c.
Using the distance formula, the distance between the points X(5,-5) and Y(-5,5) is:
Therefore, the distance between the points X(5,-5) and Y(-5,5) is about 14.1 units.