Respuesta :
Answer:
b.
[tex]V = \pi \cdot \int\limits^a_b {\left([f(x)]^2 - [g(x)]^2} \right) \, dx[/tex]
(c)
[tex]V = \pi \cdot \int\limits^3_1 {\left([arcos(y - 2)]^2 - [arcsine(x)]^2 - (-1)^2} \right) \, dx[/tex]
Step-by-step explanation:
b. The volume of solid formed is given by the washers formula as follows;
[tex]V = \pi \cdot \int\limits^a_b {\left([f(x)]^2 - [g(x)]^2} \right) \, dx[/tex]
Therefore, we have, the integral expression whose solution is the volume formed by rotating 'R', about the equations y = 2 + cos(x) and y = csc(x) in the first quadrant on the interval, 0 ≤ x ≤ π, V is given as follows;
[tex]V = \pi \cdot \int\limits^\pi_0 {\left([2 + cox(x)]^2 - [csc(x)]^2} \right) \, dx[/tex]
(c) We have;
x = arcos(y - 2), x = arcsin(1/y)
At x = 0, y = 2 + cos(0) = 3
csc(0) = ∞
At x = π, y = 2 + cos(π) = 2 + -1 = 1
csc(π) = ∞
Therefore, we get;
[tex]V = \pi \cdot \int\limits^3_1 {\left([arcos(y - 2)]^2 - [arcsine(x)]^2 - (-1)^2} \right) \, dx[/tex]
B) An equation that involves integral expressions whose solution is the volume of the solid generated when R is revolved around the x-axis is;
[tex]V = \pi \int\limits^\pi _0 ({[2 + cos(x)]^{2} - [csc(x)]^{2}}) \, dx[/tex]
C) An equation involving integral expressions whose solution is the volume of the solid generated when R is revolved around the line x= -1 is;
[tex]V = \pi \int\limits^\pi _0 ({[cos^{-1} (y - 2)]^{2} - [sin^{-1}(x)]^{2} - (-1)^{2} }) \, dx[/tex]
How to find the integral volume of solid?
B) The volume of solid formed is gotten from applying the washers formula;
[tex]V = \pi \int\limits^a_b ({[f(x)]^{2} - [g(x)]^{2}}) \, dx[/tex]
This means that the integral expression whose solution is the volume formed by rotating R about the equations y = 2 + cos(x) and y = csc(x) in the first quadrant on the interval, 0 ≤ x ≤ π, V is expressed as;
[tex]V = \pi \int\limits^\pi _0 ({[2 + cos(x)]^{2} - [csc(x)]^{2}}) \, dx[/tex]
C) From answer above, we have;
x = cos⁻¹(y - 2), x = sin⁻¹(1/y)
Now,
At x = 0; y = 2 + cos(0) = 3
csc(0) = 1/0 = ∞
Also,
At x = π; y = 2 + cos(π)
y = 2 + (-1)
y = 1
Also, csc(π) = ∞
Thus, we have;
[tex]V = \pi \int\limits^\pi _0 ({[cos^{-1} (y - 2)]^{2} - [sin^{-1}(x)]^{2} - (-1)^{2} }) \, dx[/tex]
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