# Chapter 7 Practice Exam Nicole, Valerie, Nat, Maryalice

```7.1- Nicole
7.2- Valerie, Maryalice
7.3- Nathaniel (ME!)
7.4- Yasmeen
Nicole Stevenson
7.1 Review
#1:
Question:
#2
Question:
#3
Question:
7.2 Review
Valerie
1. Question:
Find the area of the shaded region analytically: (refer to page 395 #5 for graph)
Solution:
𝐵
∫𝐴 𝑓(𝑥) − 𝑔(𝑥)𝑑𝑥
4 3
𝑥
3
1
− 5 𝑥5
4
2
2
∫−2 2𝑥 2 𝑑𝑥 − ∫−2 𝑥 4 − 2𝑥 2 𝑑𝑥
1
4
∫ 2𝑥 2 − 𝑥 4 + 2𝑥 2 𝑑𝑥
1
[3 (2)3 − 5 (2)5 ] − [3 (−2)3 − 5 (−2)5 ] =
∫ 4𝑥 2 − 𝑥 4
128
15
2. Question:
Find the area of the regions enclosed by the lines and curves: 𝑦 = 𝑥 2 − 2 𝑎𝑛𝑑 𝑦 = 2
Solution:
𝑥2 − 2 = 2
𝑥2 − 4 = 0
2
𝑥 = 2, −2 First; Find the x-intercepts
2
∫ 2 − 𝑥 2 + 2𝑑𝑥
−2
∫ 4 − 𝑥 2 𝑑𝑥 = 4𝑥 −
−2
𝑥3
3
[4(2) −
(2)3
(−2)3
32
] − [4(−2) −
]=
3
3
3
3. Question:
Find the area of the regions enclosed by the lines and curves: 𝑥 + 𝑦 2 = 0 𝑎𝑛𝑑 𝑥 + 3𝑦 2 = 2
Solution:
𝑥 = −𝑦 2
𝑥 = 2 − 3𝑦 2 First; Solve equations for x
−𝑦 2 = 2 − 3𝑦 2
1
𝑦 = 1, −1 Next; Set both equal and solve for y
∫−1[(2 − 3𝑦 2 ) − (−𝑦 2 )]𝑑𝑦
1
2
∫−1 2 − 2𝑦 2 𝑑𝑦 = 2𝑦 − 3 𝑦 3
2
2
8
[2(1) − 3 (1)3 ] − [2(−1) − 3 (−1)3 ] = 3
7.2 Two
Maryalice Weed
AP Calculus
Period 6
Section 7.2 Review
1. Find the area of the region R in the first quadrant that is bounded above by 𝑦 = √𝑥 and below
by the
x-axis and the line 𝑦 = 𝑥 − 2
Solution: 10/3
2. Find the area between the x-axis and the function 𝑦 = √9 − 𝑥 2 over the interval [-3, 3]
Solution: 5/6 or 9𝜋/2
3. Find the area of the region first quadrant bounded by the x-axis and the graphs of √𝑥 − 2 and
𝑦 = 𝑥 − 10 by subtracting the area of the triangular region from the area under the square root
curve.
Solution: 19.88
7.3 Nathaniel Kreiman
Question:
Find the volume of the solid that lies between the planes perpendicular to the x-axis at x=0 and x=4. The
cross sections perpendicular to the axis on the interval [0,4] are squares whose diagonals run from
𝑦 = −√𝑥 to 𝑦 = √𝑥
Solution:
Question:
Find the volume of the solid created by rotating the given region about the x-axis. The region is bounded
by 𝑥 + 2𝑦 = 2, the x-axis and the y-axis.
Solution:
Question:
Find the volume of the solid generated by revolving the region bounded by the lines and curves about
the x-axis. 𝑦 = 𝑥 2 + 1, 𝑦 = 𝑥 + 3.
Solution: (Following two pages.)
7.4
Question:
Solution:
Question:
Solution:
Question:
Solution:
```