Integration with Chain Rule
Practice
5.1 a) Find the area under 
.
b) Find the average y-value for 
c) Find the x-value on 
5.2 Let 
a) Find F(8)
b) Find F`(x)
is just a constant so 
This is the fundamental theorem of calculus
c) Find F`(8)
Calculate each indefinite integral.
5.3 
5.4 
5.5 
5.6 
5.7 
5.8 
5.9 
, trig identity
5.10 
Evaluate each definite integral:
5.11 
5.12 
5.13 
5.14 
5.15 
5.16 a) Find the area under 
b) Estimate the area using a trapezoidal Riemann sum with four intervals.
x |
y-left (b1) |
y-right (b2) |
Δx |
Area |
0 – 1 |
0 |
1.25 |
1 |
1.125 |
1 – 2 |
1.25 |
2 |
1 |
1.625 |
2 – 3 |
2 |
3.25 |
1 |
2.625 |
3 – 4 |
3.25 |
5 |
1 |
4.125 |
So the Riemann sum:
c) Why is the estimate an overestimate?
The graph is concave up.
