Calculus Answers

Find the volume generated by revolving the area bounded by the lines x + y = 5, x = 0

and y = 0, about x-axis.

Find the volume generated by revolving the area enclosed by the curve x2 + y2 = 9 about

the x-axis.

Find the area bounded by the curves y = x3, y = 1, and x = 0 using double integration.

A kite , at a height of 60 ft. is moving

horizontally at a rate of 5ft/sec. away from

the boy who flies it. How fast is the string

being released when 100 ft. are out.

Ship A is travelling south at the rate of 2 km/hr, at the instant that ship B, which is 32 miles south of ship A, is travelling east at rate of 4 km/hr.

a) Are they separating or approaching at the end of 2 hrs, and at what rate? b) At what time are they nearest together?

c) What is their minimum distance apart?

The sides of an equilateral triangle are increasing at the rate of 3 cm/min. Find: a) the rate of change of the perimeter.

b) the rate of change of the area when the side is 3 cm. long.

A boy is flying a kite at a height of 150 ft. If the kite moves horizontally away from the boy at the rate of 20 ft/sec, how fast is the string being paid out when the kite is 250 ft from him?

A spherical snowball with an outer layer of ice melts, so that the radius of the snowball decreases at the rate of 1/5 cm/sec. Find the rate at which the volume decreases when the diameter is 50 cm.

[R] Check the convergence of the sequence defined by 𝑢𝑛+1 = 𝑎/ 1+𝑢𝑛 where 𝑎 > 0, 𝑢1 > 0.

Test the convergence of ∑︁∞

n=1

[︂

n!2n

2

n

]︂

.

Ans: ∑︁un Converges

25. Test the convergence of ∑︁∞

n=1

[︂

n

3 + 1

2

n + 1]︂

.

Ans: ∑︁un Converges

Hint: Use vn as

1

√n

.

26. Test the convergence of

1

2 · 3 · 4

+

2

3 · 4 · 5

+

3

4 · 5 · 6

+

4

5 · 6 · 7

+ · · · .

Ans: ∑︁un Converges

Hint: Use vn as

1

n2

.

27. Test the convergence of

√

2 − 1

3

3 − 1

+

√

3 − 1

4

3 − 1

+

√

4 − 1

5

3 − 1

+ · · · .

Ans: ∑︁un Converges

Hint: Use vn as

1

n

5

2

.

28. Test the convergence of

2

1

p +

3

2

p +

4

3

p + · · · .

Ans: ∑︁un Converges if p > 2 and ∑︁un Diverges if p ≤ 2

Hint: Use vn as

1

n

p−1

.