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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
![Find the area between the curve y = x² + 12x + 27 and the x-axis from x = -3 to x = -1.
Step 1
Notice that the curve is a parabola with x-intercepts (-3
-3 ,0) and (-9, 0). Within our interval
[-3, -1], f(x) is nonnegative
nonnegative
Step 2
Therefore, our function is continuous and f(x) 2 0 on [a, b], and we can use the definite interval as follows to
measure the area under the curve.
If fis a continuous function on [a, b] and f(x) > 0 on [a, b], then the exact area between y = f(x) and the
x-axis from x = a to x = b is given by
Area = rw) dx.
9.
Therefore, set up the definite integral for our function.
+ 12x + 27
1 + 12r + 27
dx
Step 3
Now we use the Fundamental Theorem of Calculus to evaluate the integral. First integrate and write the
9.
answer in the form
(x) dx = F(x)",
where F'(x) = f(x).
| (x²+ 12x + 27) dx =
-3
-3](https://content.bartleby.com/qna-images/question/95191ecd-4227-4aa7-82d4-51509f79a5b2/73de46de-ec6b-40bf-b001-fe8ee26d1649/c5wsnm_thumbnail.png)
Transcribed Image Text:Find the area between the curve y = x² + 12x + 27 and the x-axis from x = -3 to x = -1.
Step 1
Notice that the curve is a parabola with x-intercepts (-3
-3 ,0) and (-9, 0). Within our interval
[-3, -1], f(x) is nonnegative
nonnegative
Step 2
Therefore, our function is continuous and f(x) 2 0 on [a, b], and we can use the definite interval as follows to
measure the area under the curve.
If fis a continuous function on [a, b] and f(x) > 0 on [a, b], then the exact area between y = f(x) and the
x-axis from x = a to x = b is given by
Area = rw) dx.
9.
Therefore, set up the definite integral for our function.
+ 12x + 27
1 + 12r + 27
dx
Step 3
Now we use the Fundamental Theorem of Calculus to evaluate the integral. First integrate and write the
9.
answer in the form
(x) dx = F(x)",
where F'(x) = f(x).
| (x²+ 12x + 27) dx =
-3
-3
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