Calculus 6.
7 Definite Integrals Notes
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An antiderivative of a function 𝑓 𝑥 is a function 𝐹 𝑥 whose derivative is 𝑓 𝑥 .
For example, let 𝑓 𝑥 3𝑥 and 𝑓 𝑥 6𝑥. The expression 3𝑥 is an antiderivative of
6𝑥. Why?
When we take an integral, it is taking the antiderivative of a function. The area under the
curve is represented by an antiderivative! What?!
Power Rule for finding a derivative. Antiderivative is the reverse order.
𝑓 𝑥 𝑥 𝑓 𝑥 𝑥
𝑓 𝑥 𝑛𝑥 𝑥
𝐹 𝑥
𝑛 1
Step one: Multiply by the old exponent. Step one: Add one to the exponent.
Step two: Subtract one from the exponent. Step two: Divide by the new exponent.
Given the function 𝒇 𝒙 , find the antiderivative 𝑭 𝒙 .
1. 𝑓 𝑥 𝑥 2. 𝑓 𝑥 5𝑥 1 3. 𝑓 𝑥 √𝑥 √
Sine and Cosine Integrals
sin 𝑥 𝑑𝑥 cos 𝑥 cos 𝑥 𝑑𝑥 sin 𝑥
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The Fundamental Theorem of Calculus
If 𝑓 is continuous on the interval 𝑎, 𝑏 , then the area under the curve of 𝑓 from 𝑎, 𝑏
can be represented by
𝑓 𝑥 𝑑𝑥 𝐹 𝑏 𝐹 𝑎
where 𝐹 𝑥 is the antiderivative of 𝑓.
Evaluate the definite integral. Use a calculator to check your answer.
4. 4 6𝑥 𝑑𝑥 5. √𝑥 𝑑𝑥 6. 4 sin 𝑥 𝑑𝑥
7. If 𝑓 𝑥 𝑥 2 and 𝑓 1 2, then 𝑓 3
y
𝑓 𝑥
6.7 Definite Integrals
Calculus
Practice
Find the value of the definite integral. Use a calculator to check your answer.
1. 2𝑥 4 𝑑𝑥 2. sin 𝑥 𝑥 𝑑𝑥 3. 6𝑥 8 𝑑𝑥
4. 𝑑𝑥 5. 1 𝑑𝑥 6. 2 cos 𝑥 𝑑𝑥
√
For # 7-13, use the given information to find the value of the function.
7. If 𝑓 𝑥 cos 𝑥 and 𝑓 𝜋 12, then 𝑓 8. Calculator active. If 𝑓 𝑥 sin 3𝑥 𝑒 and
𝑓 1 0.751, then 𝑓 4
9. Let 𝑓 be a differentiable function such that 𝑓 1 10. Calculator active. Let 𝑓 be a differentiable
4 and 𝑓 𝑥 6𝑥 3 . What is the value of function such that 𝑓 0 0.5 and
𝑓 3 ? 𝑓 𝑥 2 cos 𝑒𝑥 . What is the value of
𝑓 2 ?
11. Let ℎ 𝑥 be an antiderivative of 5 3𝑥 . If 12. Calculator active. Let 𝐹 𝑥 be an antiderivative
ℎ 1 3, then ℎ 2 of . If 𝐹 2 0.13, then 𝐹 5
6.7 Definite Integrals Test Prep
13. The graph of 𝑓 is shown in the figure to the right. If
𝑓 𝑥 𝑑𝑥 3.5 and 𝐹 𝑥 𝑓 𝑥 , then 𝐹 4 𝐹 0
3
2
1
1 2 3 4 5 6
(A) 6.5 (B) 1.5 (C) 2.5 (D) 5.5 (E) 4.5
14. Calculator active problem. Let 𝑓 𝑥 cos 𝑡 𝑑𝑡. At how many points in the closed interval √𝜋, √𝜋
does the instantaneous rate of change of 𝑓 equal the average rate of change of 𝑓 on that interval?
(A) Zero (B) One (C) Two (D) Three (E) Four
𝑥 1 for 𝑥 0
15. Given ℎ 𝑥 , find ℎ 𝑥 𝑑𝑥
sin 𝑥 for 𝑥 0
(A) (B) (C) (D) (E)
16. A cubic polynomial function 𝑓 is defined by 𝑓 𝑥 𝑥 𝑎𝑥 𝑏𝑥 𝑐, where 𝑎, 𝑏, and 𝑐 are constants.
The function 𝑓 has a local minimum at 𝑥 2, and the graph of 𝑓 has a point of inflection at 𝑥 5. If
𝑓 𝑥 𝑑𝑥 , what is the value of 𝑐?
