Math 201: Integrals (5.2, 5.3) Day 53
Recall the definition of the definite integral:
Zb
a
f(x)dx = lim
N→∞
RN= lim
N→∞
∆x
N
X
j=1
f(a+j∆x)
.
Use this definition to compute the following integrals.
(1) Z2
0
2x−3dx
(2) Z3
0
4x2dx
The Fundamental Theorem of Calculus (FTC) tells us an easier way to compute definite integrals:
Zb
a
f(x)dx =F(b)−F(a)
where F(x) is any antiderivative of f(x) (that is F0(x) = f(x)). Use the FTC to compute the
following integrals.
(1) Z3
0
4x2dx
(2) Z7
2
3x6dx
(3) Z2π
0
sin(x)dx
(4) Z1
0
1
x2+ 1 dx
(5) Z4
1
ex+ sec2(x)dx
Use the properties of integrals and
Z6
2
f(x)dx = 5 Z6
2
g(x)dx = 2
to compute the integral Z6
2
2f(x)−4g(x)+5 dx.
1