Unformatted text preview: Example
Evaluate the deﬁnite integral
3 2x dx
0 by interpreting it as an area of a particular region.
Solution: The integrand is 2x, and since the line y = 2x does not go below the x-axis for
0 ≤ x ≤ 3, this deﬁnite integrand can be interpreted as the area under the line y = 2x and
above the x-axis, and between the vertical lines x = 0 and x = 3.
The region is a right triangle with height h given by the height of the line y = 2x at the right
endpoint x = 3, i.e., h = 2 · 3, and with base b given by the length of the integration interval,
or b = 3 − 0 = 3. So the area is
bh = 3 · 6 = 9
square units. It follows that
3 2x dx = 9.
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