Chapter 6
Using Definite Integrals
6.1 Using Definite Integrals to Find Area and Length
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we use definite integrals to measure the area between two curves?
• How do we decide whether to integrate with respect to x or with respect to y when
we try to find the area of a region?
• How can a definite integral be used to measure the length of a curve?
Introduction
Early on in our work with the definite integral, we learned that if we have a nonnegative
velocity function, v, for an object moving along an axis, the area under the velocity
function between a and b tells us the distance the object traveled on that time interval.
Moreover, based on the definition of the definite integral, that area is given precisely by
b
a
v(t) dt. Indeed, for any nonnegative function f on an interval [a, b], we know that
b
a
f (x) dx measures the area bounded by the curve and the x-axis between x = a and
x = b.
Through our upcoming work in the present section and chapter, we will explore how
definite integrals can be used to represent a variety of different physically important
properties. In Preview Activity 6.1, we begin this investigation by seeing how a single
definite integral may be used to represent the area between two curves.
333
Using Definite Integrals
6.1 Using Definite Integrals to Find Area and Length
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we use definite integrals to measure the area between two curves?
• How do we decide whether to integrate with respect to x or with respect to y when
we try to find the area of a region?
• How can a definite integral be used to measure the length of a curve?
Introduction
Early on in our work with the definite integral, we learned that if we have a nonnegative
velocity function, v, for an object moving along an axis, the area under the velocity
function between a and b tells us the distance the object traveled on that time interval.
Moreover, based on the definition of the definite integral, that area is given precisely by
b
a
v(t) dt. Indeed, for any nonnegative function f on an interval [a, b], we know that
b
a
f (x) dx measures the area bounded by the curve and the x-axis between x = a and
x = b.
Through our upcoming work in the present section and chapter, we will explore how
definite integrals can be used to represent a variety of different physically important
properties. In Preview Activity 6.1, we begin this investigation by seeing how a single
definite integral may be used to represent the area between two curves.
333