Math

Question 1

Area Between Curves Formula

Answer 1

Question 2

How do you find the bounds if they are not given?

Answer 2

Set f(x) = g(x) and solve for x

Question 3

Find the area of the shaded region

Answer 3

343/6

Question 4

How do you calculate area between curves in terms of y

Answer 4

Integral of Right function - Left function

Question 5

How do you calculate area between curves in terms of x

Answer 5

Integral of Top function - Bottom function

Question 6

If the curves intersect at multiple points…

Answer 6

Split the equation into multiple integrals and add them together

Question 7

Find the area of the region bounded by y = sec^2(x) and y = 8cos(x) from -pi/3 to pi/3

Answer 7

6*sqrt3

Question 8

Find the area of the region bounded by y = x^2 and y = 4x-x^2

Answer 8

8/3

Question 9

Set up the integral for the area of a region bounded by y = cos(pi*x) and y = 4x^2 - 1

Answer 9

A = 2 * integral from 0 to 1/2 of cos(pi*x)-(4x^2 - 1)dx

Question 10

Find the area of the region bounded by x = y^4, y = sqrt(2-x) and y=0

Answer 10

22/15

Question 11

Find the volume of a solid obtained by rotating the region bounded by y = x^3, y-axis, x = 0 and y = 8, about the y-axis

Answer 11

96pi/5

Question 12

Find the volume of a region bounded by y=x and y=x^2 rotated about the x-axis

Answer 12

2pi/15

Question 13

Find the volume of a region bounded by y=x and y=x^2 rotated around y = 2

Answer 13

8pi/15

Question 14

Find the volume of a square pyramid with base length “l” and height “h”

Answer 14

1/3(l^2*h)

Question 15

Volume of shell formula

Answer 15

Question 16

What method should be used if you’re rotating around a vertical axis?

Answer 16

Variable x, use shell method; variable y, use disk/washer method

Question 17

What method should be used if you’re rotating around a horizontal axis?

Answer 17

Variable x, use disk/washer method; variable y, use shell method

Question 18

Find the volume of the region formed by rotating y = x and y = x^2 around the y-axis.

Answer 18

pi/6

Question 19

Find the volume of the solid generated by rotating the region bounded by y = sqrt(x) and y = 0.

Answer 19

pi/2

Question 20

Find the volume of the solid generated by rotating the region bounded by the x-axis and y = x - x^2 about x = 2.

Answer 20

pi/2

Question 21

Calculus definition of work

Answer 21

Question 22

A 200 lb. cable is hanging off the roof of a building. The cable is 100 ft. long (the building is taller). How much work is done lifting the cable to the roof.

Answer 22

1,000 ft. lbs. or 1355.8 joules

Question 23

A force of 40 N is regained to hold a spring with a natural length of 10 cm to a length of 15 cm. How much work is needed to stretch the spring from 15 cm to 18 cm?

Answer 23

1.565 joules

Question 24

Limit definition of average value of a function

Answer 24

Question 25

Average value integral formula

Answer 25

Question 26

Find the average value of f(x) = 1 + x^2 from a = -1 to b = 2

Answer 26

2

Question 27

If f(x) is continuous on [a, b] then there exists a a<=c<=b such that f(c) = f,avg

Answer 27

Mean value theorem for integrals

Question 28

Find the value c that satisfies the mean value theorem for f(x) = 1 + x^2 from x = -1 to x = 2

Answer 28

c = -1, 1

Question 29

Arc length formula

Answer 29

Question 30

Surface area formula

Answer 30

Question 31

Function that consists of only natural numbers

Answer 31

Sequence

Question 32

Sequence formula

Answer 32

sn = s1 + d(n - 1) n = next term d = coefficient applied to each term

Question 33

Sum of the elements of a sequence

Answer 33

Series

Question 34

A series is ________ if the sequence of partial sums is a _________ sequence. A series is _________ if it is not __________.

Answer 34

Convergent, convergent, divergent, convergent

Question 35

If the limit of a series as n approaches infinity does not equal 0, the series ______

Answer 35

Diverges

Question 36

Sum of a geometric series

Answer 36

S = a/(1 - r)

Question 37

Geometric series formula

Answer 37

Question 38

A series converges if…

Answer 38

The absolute value of r is less than 1

Question 39

What is the term for a series that can be written as tn = an - an+1

Answer 39

Telescoping series

Question 40

Nth partial sum formula

Answer 40

Question 41

A series whose terms alternate between positive and negative

Answer 41

Alternating series

Question 42

What are the 2 conditions of the alternating series test?

Answer 42

Question 43

Alternating Series Estimation Theorem

Answer 43

Question 44

Comparison Test Conditions

Answer 44

Question 45

Answer 45

Based on the comparison test, the series converges

Question 46

Root Test Conditions

Answer 46

NOTE: Not commonly used

Question 47

Ratio test conditions

Answer 47

Question 48

If the absolute value of a series converges, but not the original. Then it is said to be

Answer 48

Absolutely convergent

Question 49

If the absolute value of a series diverges, but the original converges, then it is said to be…

Answer 49

Conditionally convergent

Question 50

What test is used given these conditions?

Answer 50

Divergence test

Question 51

What test is used given these conditions?

Answer 51

Geometric series test

Question 52

What test is used given these conditions?

Answer 52

P-series test

Question 53

What test is used given these conditions?

Answer 53

Direct comparison test

Question 54

What test is used given these conditions?

Answer 54

Limit comparison test

Question 55

What test is used given these conditions?

Answer 55

Alternating series test

Question 56

What test is used given these conditions?

Answer 56

Alternating series test

Question 57

What test is used given these conditions?

Answer 57

Root test

Question 58

What test is used given these conditions?

Answer 58

Ratio test

Question 59

What test is used given these conditions?

Answer 59

Telescoping series test

Question 60

What test is used given these conditions?

Answer 60

Integral test