Rutgers University Calculus Zoom Proctored Final Exam thank you for taking on my assignment on such a busy time for you.Please do it as the same format you

Rutgers University Calculus Zoom Proctored Final Exam thank you for taking on my assignment on such a busy time for you.Please do it as the same format you did last time because I have to right then on paper. If you can please get it done by 6 it’ll be amazing. And just like last time if you would like you can watch videos and get an idea. Name: ___________________
Prof. Yeung
Zoom Proctored Final Exam
Date: ____________________
MA 120 Summer 2020
Direction: Please show all your work for each of the following problems in order to receive full credits.
1. Use a simple area formula from geometry to find the area under function ( ) that gives the area
between the function ( ) = 5 − 8 and the interval [ , ] = [−2, ] (12 points)
2. Evaluate the following integral: (15 points)
+5
a) ∫ 5
√
b) ∫( + 2)√
3
c) ∫ ( + 2 )
Cos
3. Evaluate the integrals using appropriate substitutions. (18 points)
8
a) ∫ 7 √ 8 − 2

b) ∫ 2
√4 +2
c) ∫ tan7 7 sec 2 7
4. Applying sigma notation and related theorem: (12 points)
a) Evaluate

2 1 2
∑( ) ( )

=1
b) Evaluate

∑ (5 +
=1
9 1
)

5. Given ( ) = √3 + + 4. Use the definition of 4.4.3 to evaluate each of the area in the range [0, 6] if
= 4. (18 points)
a) Find the area under the curve when ∗ is the left endpoint of the subinterval.
b) Find the area under the curve when ∗ is the midpoints of the subinterval.
c) Find the area under the curve when ∗ is the right endpoint of the subinterval.
6. Evaluate the integrals using the Fundamental Theorem of Calculus and (if necessary) properties of the
definite integral. (18 points)
4
2,
>2
a) ∫−1 ( ) if ( ) = {
2 − , ≤ 2
1

b) ∫04 ( + 9sec 2 )
2
c) ∫
8
sin
7. Find all value of ∗ in the stated interval that satisfy the Mean-Value Theorem for Integrals (4.6.2).

Given ( ) = 8 sin ; [ , 2 ]. Hint: Use your answer from question # 6 part c) to help you satisfy
4.6.2 Therorem. (7 points)
2

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