Evaluate the following definite integrals as limits of sums:
Question 1.
Solution:
We have,
I =
We know,
, where h = Here a = 0, b = 3 and f(x) = x + 4.
=> h = 3/n
=> nh = 3
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 12 +
=
Therefore, the value of
as limit of sum is .
Question 2.
Solution:
We have,
I =
We know,
, where h = Here a = 0, b = 2 and f(x) = x + 3.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 6 + 2
= 8
Therefore, the value of
as limit of sum is 8.
Question 3.
Solution:
We have,
I =
We know,
, where h = Here a = 1, b = 3 and f(x) = 3x − 2.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 2 + 6
= 8
Therefore, the value of
as limit of sum is 8.
Question 4.
Solution:
We have,
I =
We know,
, where h = Here a = −1, b = 1 and f(x) = x + 3.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 4 + 2
= 6
Therefore, the value of
as limit of sum is 6.
Question 5.
Solution:
We have,
I =
We know,
, where h = Here a = 0, b = 5 and f(x) = x + 1.
=> h = 5/n
=> nh = 5
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 5 +
=
Therefore, the value of
as limit of sum is .
Question 6.
Solution:
We have,
I =
We know,
, where Here a = 1, b = 3 and f(x) = 2x + 3.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 10 + 4
= 14
Therefore, the value of
as limit of sum is 14.
Question 7.
Solution:
We have,
I =
We know,
, where h = Here a = 3, b = 5 and f(x) = 2 − x.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= –2 – 2
= –4
Therefore, the value of
as limit of sum is –4.
Question 8.
Solution:
We have,
I =
We know,
, where h = Here a = 0, b = 2 and f(x) = x2 + 1.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
=
Therefore, the value of
as limit of sum is .
Question 9.
Solution:
We have,
I =
We know,
, where h = Here a = 1, b = 2 and f(x) = x2.
=> h = 1/n
=> nh = 1
So, we get,
I =
=
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 1 + 1 +
= 1 + 1 +
=
Therefore, the value of
as limit of sum is .
Question 10.
Solution:
We have,
I =
We know,
, where h = Here a = 2, b = 3 and f(x) = 2x2 + 1.
=> h = 1/n
=> nh = 1
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 9 + 4 +
=
Therefore, the value of
as limit of sum is .
Question 11.
Solution:
We have,
I =
We know,
, where h = Here a = 1, b = 2 and f(x) = x2 − 1.
=> h = 1/n
=> nh = 1
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 1 +
= 1 +
=
Therefore, the value of
as limit of sum is .