1
More Properties of Integrals- HW Problems
1. Evaluate the following definite integrals using symmetry (i.e. 𝑓
odd/even) where applicable.
𝜋
2
𝑎. ∫ (𝑥𝑐𝑜𝑠(𝑥 ) + cos(𝑥 ))𝑑𝑥
−
𝜋
2
𝜋
𝑏. ∫ 𝑥 2 sin(𝑥 ) 𝑑𝑥
6
−
𝜋
6
1
𝑐. ∫−1(2 − |𝑥 |)𝑑𝑥
Find the average value of the function on the given interval.
2. 𝑓(𝑥 ) = 𝑥 2 − 9 on [−3,3]
3. 𝐺 (𝑥 ) = sin(𝑥 ) on [0, 𝜋]
𝜋 𝜋
4. ℎ(𝑥 ) = cos(𝑥 ) on [− 2 , 2 ]
5. The profit in thousands of dollars over time of a firm’s new product
is given by 𝑃(𝑡) = 4(−𝑡 2 + 20𝑡 + 20), where 𝑡 is in months. Find the
average profit over the first 6 months.
Find all values of 𝑐 guaranteed by the mean value theorem for
integrals.
1
6. 𝑓(𝑥 ) = 𝑥 2 on [1,4].
7. 𝑓(𝑥 ) = −3𝑥 2 + 12𝑥 + 2 on [0,4].
More Properties of Integrals- HW Problems
1. Evaluate the following definite integrals using symmetry (i.e. 𝑓
odd/even) where applicable.
𝜋
2
𝑎. ∫ (𝑥𝑐𝑜𝑠(𝑥 ) + cos(𝑥 ))𝑑𝑥
−
𝜋
2
𝜋
𝑏. ∫ 𝑥 2 sin(𝑥 ) 𝑑𝑥
6
−
𝜋
6
1
𝑐. ∫−1(2 − |𝑥 |)𝑑𝑥
Find the average value of the function on the given interval.
2. 𝑓(𝑥 ) = 𝑥 2 − 9 on [−3,3]
3. 𝐺 (𝑥 ) = sin(𝑥 ) on [0, 𝜋]
𝜋 𝜋
4. ℎ(𝑥 ) = cos(𝑥 ) on [− 2 , 2 ]
5. The profit in thousands of dollars over time of a firm’s new product
is given by 𝑃(𝑡) = 4(−𝑡 2 + 20𝑡 + 20), where 𝑡 is in months. Find the
average profit over the first 6 months.
Find all values of 𝑐 guaranteed by the mean value theorem for
integrals.
1
6. 𝑓(𝑥 ) = 𝑥 2 on [1,4].
7. 𝑓(𝑥 ) = −3𝑥 2 + 12𝑥 + 2 on [0,4].
