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CBSE Class 12 Math 2020 Delhi Set 1 Solved Paper

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Question : 34
Total: 36
Using integration find the area of the region bounded between the two circles x2+y2=9 and (x−3)2+y2=9.
OR
Evaluate the following integral as the limit of sums
4
∫
1
(x2−x)dx.
Solution:     👈: Video Solution

OR

Let I=
4
∫
1
(x2−x)‌dx
We know
b
∫
a
f(x)‌dx=
lim
n→∞
h[f(a)+f(a+h)+f(a+2h)+...+f(a+(n−1)h)] ,
As n→∞,h→0⇒nh=b−a=4−1=3
∴
b
∫
a
f(x)‌dx=
lim
n→∞
h‌
n−1
∑
r=0
f(a+rh)‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
Here f(x)=x2−x,a=1,b=4 .
∴f(a+rh)=(a+rh)2−(a+rh)
⇒f(1+rh)=(1+rh)2−(1+rh)

By using (i),
4
∫
1
(x2−x)‌dx=
lim
n→∞
h‌
n−1
∑
r=0
[r2h2+rh]
⇒I=
lim
n→∞
h{h2‌
n−1
∑
r=0
r2+h‌
n−1
∑
r=0
r}
⇒I=
lim
n→∞
h{h2×‌
n(n−1)(2n−1)
6
+h‌
n(n−1)
2
}
⇒I=
lim
n→∞
{‌
nh(nh−h)(2nh−h)
6
+‌
nh(nh−h)
2
}
⇒I=‌
3(3−0)(6−0)
6
+‌
3(3−0)
2

⇒I=9+‌
9
2
=‌
27
2
.
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