Find the area of the infinite region between the curve y^2(2a-x)=x^3 and its asymptote.
Answers
Answer:
THIS IS YOUR ANSWER
Step-by-step explanation:
f(1) = 2, f
0
is continuous and R 4
1
f
0
(x)dx = 17. What is the value of f(4)?.
1. Length of the curves
(a) Determine the perimeter of one loop of the curve 6ay2 = x(x − 2a)
2
[Ans. √
8a
3
]
(b) Calculate the distance traveled by the particle P(x, y) after 4 minutes, if the position at any
time is given by x =
t
2
2
, y =
1
3
(2t + 1)3/2
. [Ans. 12]
(c) Find the perimeter of the curve r = a(cos θ + sin θ) [Ans. √
2πa]
(d) Determine the perimeter of the curve r = a sin3
(
θ
3
) [Ans. 3πa
2
]
(e) Find the length of the arc of the parabola r =
2a
(1+cos θ)
cut-off by its latus rectum.
[Ans. (√
2 + ln(1 + √
2))2a]
(f) Find the length of the astroid x = a cos3
t, y = a sin3
t. [Ans. 6a]
2. Area the curves
(a) Determine the area between the cubic y = x
3 and the parabola y = 4x
2
. [Ans. 64
3
]
(b) Calculate the area between the curve y
2
(a + x) = (a − x)
3 and its asymptotes. [Ans. 3πa2
]
(c) Find the whole area bounded by the four infinite branches of the tractrix:
x = a cost +
1
2
a ln tan2 t
2
, y = a sin t. [Ans. πa2
]
(d) Find the whole area of the curves (i) r = a cos nθ (ii) r = a sin nθ (iii) r = a cos 3θ + b sin 3θ.
[Ans. (i) πa2
4n
(ii) πa2
4n
, (iii) π(a
2+b
2
)
4
]
(e) Let P Q be the common tangent to the two loops of the lemniscate r
2 = a
2
cos 2θ with pole O.
Find the area bounded by the line P Q and the arcs OP and OQ of the curve.
[Ans. a
2
8
(3√
3 − 4)]
(f) Compute the area bounded by the x-axis and an arc of the cycloid x = a(t−sin t),y = a(1−cost).
[Ans. 3πa2
]
3. Volume and Surfaces of Solid of Revolution
(a) Find the volume of the solid generated by the revolution of an arc of the catenary y = c cosh(x/c)
about the x-axis. [Ans. πc2
2
(x +
c
2
sinh(2x/c))]
(b) Determine the volume of solid generated by revolving the plane area bounded by y
2 = 4x and
x = 4 about the line x = 4. [Ans. 1024
15 π]
(c) Find the volume of the solid generated by revolving the smaller area bounded by the circle
x
2 + y
2 = 2 and semicubical parabola y
3 = x
2 about the x-axis [Ans. 52
21π]
(d) Determine the volume of solid of revolution generated by revolving the curve whose parametric
equation are x = 2t + 3, y = 4t
2 − 9 about the x-axis for t1 = −
3
2
, t2 =
3
2
. [Ans. 1296π]
(e) The arc of the cardioid r = a(1 + cos θ) included between θ = −π/2 and θ = π/2 is rotated
about the line θ = π/2. Find the volume of the solid of revolution. [Ans. πa3
4
(16 + 5π)]
(f) Find the volume of the solid generated by the revolution of the catenary y =
a
2
(e
x
a +e
−x
a ) about
the x-axis between x = 0 and x =
π
2
. [Ans. πa3
8
(e
2b/a − e
−2b/a) + πa2
b
2
]