Definite integral as an area: Example 2
Find the value of the integral
Z
4
(4 − x) dx.
0
A. 8
B. 4
C. 2
D. 1
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
1/5
Definite integral as an area: Example 3
Find the value of the integral
Z
2π
sin x dx.
0
A. 1
B. 0
C. 2π
D. −2π
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
2/5
Approximating areas under curves
For an arbitrary function f , how does one compute
Math 105 (Section 204)
Integral Calculus – Definite integrals
Rb
a
f (x) dx?
2011W T2
3/5
Approximating areas under curves
For an arbitrary function f , how does one compute
Rb
a
f (x) dx?
Riemann sum
Given a function y = f (x) on the interval [a, b], its Riemann sum Rn (f ) is
an expression of the form
Rn (f ) =
n
X
k=1
f (xk )
b−a
,
n
b−a
where xk is some point in the subinterval Ik = [a + (k − 1) b−a
n , a + k n ].
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
3/5
Approximating areas under curves
For an arbitrary function f , how does one compute
Rb
a
f (x) dx?
Riemann sum
Given a function y = f (x) on the interval [a, b], its Riemann sum Rn (f ) is
an expression of the form
Rn (f ) =
n
X
k=1
f (xk )
b−a
,
n
b−a
where xk is some point in the subinterval Ik = [a + (k − 1) b−a
n , a + k n ].
The value of Rn (f ) depends on the choice of xk within Ik .
If xk = left (resply right) endpoint of Ik for all k, then Rn (f ) is called
the left (resply right) Riemann sum.
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
3/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
4/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
On each Ik , replace f by a constant function whose value on the
entire subinterval is f (xk ).
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
4/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
On each Ik , replace f by a constant function whose value on the
entire subinterval is f (xk ).
Get a step function, i.e., a piecewise constant curve that
approximates f .
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
4/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
On each Ik , replace f by a constant function whose value on the
entire subinterval is f (xk ).
Get a step function, i.e., a piecewise constant curve that
approximates f .
The Riemann sum Rn (f ) is the area under this “approximating”
curve, and as such an approximation of the area under f (see
accompanying diagram).
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
4/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
On each Ik , replace f by a constant function whose value on the
entire subinterval is f (xk ).
Get a step function, i.e., a piecewise constant curve that
approximates f .
The Riemann sum Rn (f ) is the area under this “approximating”
curve, and as such an approximation of the area under f (see
accompanying diagram).
Clearly the larger n is, the better the approximation.
Math 105 (Section 204)
Integral Calculus – Definite integrals
2011W T2
4/5
What does a Riemann sum represent?
Partition [a, b] into n equal subintervals Ik , 1 ≤ k ≤ n.
On each Ik , replace f by a constant function whose value on the
entire subinterval is f (xk ).
Get a step function, i.e., a piecewise constant curve that
approximates f .
The Riemann sum Rn (f ) is the area under this “approximating”
curve, and as such an approximation of the area under f (see
accompanying diagram).
Clearly the larger n is, the better the approximation.
Relation between Riemann sums and the definite integral
Z
b
f (x) dx = lim Rn (f ) = lim
a
Math 105 (Section 204)
n→∞
n→∞
n
X
k=1
Integral Calculus – Definite integrals
f (xk )
b−a
.
n
2011W T2
4/5
An example
Which one of the definite integrals below equals the limit
lim
n→∞
A.
Z
Z
Z
Z
1
Math 105 (Section 204)
?
1
x
dx
1 + x2
1
1
dx
1 + x2
2
x2
dx
1 + x2
0
D.
k2
n2
1
dx
1 + x2
0
C.
1+
2
1
B.
k
n2
Integral Calculus – Definite integrals
2011W T2
5/5