# Class 12 RD Sharma Solutions – Chapter 20 Definite Integrals – Exercise 20.5 | Set 1

**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 ofas 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 ofas 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 ofas 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 ofas 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 ofas 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 ofas 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 ofas limit of sum is –4.

**Question 8. **

**Solution:**

We have,

I =

We know,

, where h =

Here a = 0, b = 2 and f(x) = x

^{2}+ 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 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 ofas limit of sum is.

**Question 10. **

**Solution:**

We have,

I =

We know,

, where h =

Here a = 2, b = 3 and f(x) = 2x

^{2}+ 1.=> h = 1/n

=> nh = 1

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 9 + 4 +

=

Therefore, the value ofas limit of sum is.

**Question 11. **

**Solution:**

We have,

I =

We know,

, where h =

Here a = 1, b = 2 and f(x) = x

^{2}− 1.=> h = 1/n

=> nh = 1

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 1 +

= 1 +

=

Therefore, the value ofas limit of sum is.

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