UNIT 1

Question 1-: Evaluate

Solution:

Let

…

By L – Hospital rule,

Question-2: Evaluate

Solution:

Let

…

By L – Hospital rule

…

…

Question-3: Find the value of a, b if

Solution:

Let

…

By L – Hospital rule

…

…

… (1)

…

But

From equation (1)

Question-4: Evaluate

Solution:

Let

…

Taking log on both sides,

…

By L – Hospital rule,

i.e.

Question-5: find out the integral is convergent or divergent. Find the value in case of convergent.

Sol. Here we will convert the integral into limit ,

=

=

=

= ∞

As we can see , here limit does not exist. i.e. that is infinity.

So we can say that the given integral is divergent.

Question-6: find out the integral is convergent or divergent. Find the value in case of convergent.

Sol. Covert to the limit ,

=

=

=

Again the limit does not exist that means the integral is divergent

Question-7: find out the integral is convergent or divergent. Find the value in case of convergent.

Sol. As we see, the given is integrand is not continuous at x = 0 , we will split the integral,

= +

We will check one by one whether the integrals are convergent or divergent,

As we found that, integral is divergent

We don’t need to check for the second one.

Question-8: Find γ(-½)

Solution: (-½) + 1 = ½

γ(-1/2) = γ(-½ + 1) / (-½)

= - 2 γ(1/2 )

= - 2 π

Question-9: . Show that

Solution : =

=

= ) .......................

=

=

Question-10: ): Evaluate I =

Solution:

= 2 π/3

Question-11: Evaluate

Solution :Let

Put or ,,

When,;,

o | ||

1 | 0 |

Also

Question- 12: find the area under the curves where y = x and y = x + 1, x = 2 and y-axis.

Sol. When we draw the graph, curve does not meet, but depend on two vertical lines,

Here boundary is [0 , 2]

Area under the curve,

A =

So that the area under the curve is 3.2092 unit square.

Question-13: find the average value of the function f(x) = x³ over the interval[0,1].

Sol. We know that

f( avg.) =

= = = 1 / 4

Question-14: A force of 1200 N compresses a spring from its natural length of 18 cm to a length of 16 cm. How much work is done in compressing it from 16 cm to 14 cm?

Sol. Here,

F = kx

So that,

1200 = 2k

K = 600 N/cm

In that case,

F = 600x

We know that,

W =

W = , which gives

W = 3600 N.cm

Question-15: Find the volume of the solid of revolution generated by rotating the region between the graph of f(x) = √x over the interval [1,4] around x-axis.

Sol. The graph of the function will look like as follow,

On rotation it will make circle ( cross-section)

We know that,

V =

= = , which gives

=

The volume is

Question-16: suppose a wire hanging on two poles follows the curve,

f(x) = a cosh(x/a)

Find the length of the wire.

Sol. We will find the first derivative of f(x),

f’(x) = sinh(x/a)

The curve of f(x) will look like,

Hence the curve os symmetric, we will measure the length of one side first,

The limits on one side will be, 0 to b

We know that

Length of Arc =

Put f’(x) = sinh(x/a), we get

Length of Arc =

Use the identity,

=

=

We get on solving,

a sinh(b/a)

On both sides, the length of the curve will be,

2 a sinh(b/a)