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)
