6.1. USING DEFINITE INTEGRALS TO FIND AREA AND LENGTH
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the region whose area is being found, (iii) draw and label a representative slice, and (iv)
state the area of the representative slice. Then, state a definite integral whose value is
the exact area of the region, and evaluate the integral to find the numeric value of the
region’s area. Note well: At the step where you draw a representative slice, you need
to make a choice about whether to slice vertically or horizontally.
(a) The finite region bounded by x = y 2 and x = 6 − 2y 2 .
(b) The finite region bounded by x = 1 − y 2 and x = 2 − 2y 2 .
(c) The area bounded by the x-axis, y = x 2 , and y = 2 − x.
(d) The finite regions between the curves x = y 2 − 2y and y = x.
⊳
Finding the length of a curve
In addition to being able to use definite integrals to find the areas of certain geometric
regions, we can also use the definite integral to find the length of a portion of a curve. We
use the same fundamental principle: we take a curve whose length we cannot easily find,
and slice it up into small pieces whose lengths we can easily approximate. In particular, we
take a given curve and subdivide it into small approximating line segments, as shown at
left in Figure 6.5. To see how we find such a definite integral that measures arc length on
x
y
f
x 0
x 1
x 2
x 3
△x
△y
h
L slice
Figure 6.5: At left, a continuous function y = f (x) whose length we seek on the interval
a = x 0 to b = x 3 . At right, a close up view of a portion of the curve.
the curve y = f (x) from x = a to x = b, we think about the portion of length, L slice , that
lies along the curve on a small interval of length △x, and estimate the value of Lslice using
a well-chosen triangle. In particular, if we consider the right triangle with legs parallel to
the coordinate axes and hypotenuse connecting two points on the curve, as seen at right
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the region whose area is being found, (iii) draw and label a representative slice, and (iv)
state the area of the representative slice. Then, state a definite integral whose value is
the exact area of the region, and evaluate the integral to find the numeric value of the
region’s area. Note well: At the step where you draw a representative slice, you need
to make a choice about whether to slice vertically or horizontally.
(a) The finite region bounded by x = y 2 and x = 6 − 2y 2 .
(b) The finite region bounded by x = 1 − y 2 and x = 2 − 2y 2 .
(c) The area bounded by the x-axis, y = x 2 , and y = 2 − x.
(d) The finite regions between the curves x = y 2 − 2y and y = x.
⊳
Finding the length of a curve
In addition to being able to use definite integrals to find the areas of certain geometric
regions, we can also use the definite integral to find the length of a portion of a curve. We
use the same fundamental principle: we take a curve whose length we cannot easily find,
and slice it up into small pieces whose lengths we can easily approximate. In particular, we
take a given curve and subdivide it into small approximating line segments, as shown at
left in Figure 6.5. To see how we find such a definite integral that measures arc length on
x
y
f
x 0
x 1
x 2
x 3
△x
△y
h
L slice
Figure 6.5: At left, a continuous function y = f (x) whose length we seek on the interval
a = x 0 to b = x 3 . At right, a close up view of a portion of the curve.
the curve y = f (x) from x = a to x = b, we think about the portion of length, L slice , that
lies along the curve on a small interval of length △x, and estimate the value of Lslice using
a well-chosen triangle. In particular, if we consider the right triangle with legs parallel to
the coordinate axes and hypotenuse connecting two points on the curve, as seen at right