Integration with limits

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
∫ 3x
2
dxnolimits
1
3
= x
3
[ ]
1
3
= 3
3
1
3
= 26
+c
+c – c
Integrate
Substitute
Subtract
–

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

5. ∫ 8x
3
dx
1
2
= 2x
4
[ ]
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
= 4x
2
[ ]
1
5
= 4(5)
2
– 4(1)
2
= 96
Ex 1.

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

9. ∫ (3x
2
+ 4) dx
1
3
= x
3
+ 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
= 3x
2
+ 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
= x
2
+ 6x[ ]
o
a
= a
2
+ 6a – ( (0)
2
+ 6(0) )
Nasty
= 27 Quadratic Equation
a
2
+ 6a – 27 = 0
(a + 9)(a – 3) = 0
a = -9 or a = 3