Integration with Limits

1. Block 2
Integration with Limits

2. What is to be learned?
• What these limit things are and how they
work

3. Introducing Limits
∫ 3x2
dx no limits
1
3
= x3
[ ]
1
3
= 33
13
= 26
+c
+c – c
Integrate
Substitute
Subtract
–

4. ∫ 6x dx
1
4
= 3x2
[ ]
1
4
= 3(4)2
– 3(1)2
= 45

5. ∫ 8x3
dx
1
2
= 2x4
[ ]
1
2
= 2(2)4
– 2(1)4
= 30

6. Integrals With Limits
Have wee numbers on them
Integrate as normal
Substitute wee numbers, then subtract
No need for + c

7. ∫ 8x dx
1
5
= 4x2
[ ]
1
5
= 4(5)2
– 4(1)2
= 96
Ex 1.

8. ∫ (4x + 2) dx
1
3
= 2x2
+ 2x[ ]
1
3
= 2(3)2
+ 2(3)
= 24 – 4
– ( 2(1)2
+ 2(1) )
= 20
subtract all of this

9. ∫ (3x2
+ 4) dx
1
3
= x3
+ 4x[ ]
1
3
= (3)3
+ 4(3)
= 39 – 5
– ( (1)3
+ 4(1) )
= 34
subtract all of this
Ex 2.

10. ∫ (6x + 5) dx
1
2
= 3x2
+ 5x[ ]
1
2
= 3(2)2
+ 5(2)
= 22 – 8
– ( 3(1)2
+ 5(1) )
= 14
subtract all of this
Key Question

11. ∫ (2x + 6) dx = 27 Find a (a > 0)
0
a
= x2
+ 6x[ ]
o
a
= a2
+ 6a – ( (0)2
+ 6(0) )
Nasty
= 27 Quadratic Equation
a2
+ 6a – 27 = 0
(a + 9)(a – 3) = 0
a = -9 or a = 3