Class 12 RD Sharma Solutions – Chapter 20 Definite Integrals – Exercise 20.5 | Set 3
Last Updated :
20 May, 2021
Evaluate the following definite integrals as limits of sums:
Question 23.
Solution:
We have,
I =
We know,
, where h =
Here a = 0, b = 4 and f(x) = x + e2x.
=> h = 4/n
=> nh = 4
So, we get,
I =
=
=
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 24.
Solution:
We have,
I =
We know,
, where h =
Here a = 0, b = 2 and f(x) = x2 + x.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 25.
Solution:
We have,
I =
We know,
, where h =
Here a = 0, b = 2 and f(x) = x2 + 2x + 1.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 26.
Solution:
We have,
I =
We know,
, where h =
Here a = 0, b = 3 and f(x) = 2x2 + 3x + 5.
=> h = 3/n
=> nh = 3
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 15 + 18 +
=
Therefore, the value ofas limit of sum is.
Question 27.
Solution:
We have,
I =
We know,
, where h =
Here a = a, b = b and f(x) = x.
=> h =
=> nh = b − a
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 28.
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 ofas limit of sum is.
Question 29.
Solution:
We have,
I =
We know,
, where h =
Here a = 2, b = 3 and f(x) = x2.
=> h = 1/n
=> nh = 1
So, we get,
I =
=
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 30.
Solution:
We have,
I =
We know,
, where h =
Here a = 1, b = 3 and f(x) = x2 + x.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 31.
Solution:
We have,
I =
We know,
, where h =
Here a = 0, b = 2 and f(x) = x2 − x.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
=
=
Therefore, the value ofas limit of sum is.
Question 32.
Solution:
We have,
I =
We know,
, where h =
Here a = 1, b = 3 and f(x) = 2x2 + 5x.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 14 + 18 +
=
Therefore, the value ofas limit of sum is.
Question 33.
Solution:
We have,
I =
We know,
, where h =
Here a = 1, b = 3 and f(x) = 3x2 + 1.
=> h = 2/n
=> nh = 2
So, we get,
I =
=
=
=
Now if h −> 0, then n −> ∞. So, we have,
=
=
= 8 + 12 + 8
= 28
Therefore, the value ofas limit of sum is 28.
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