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Nicholas > Math > Flashcards

Math Flashcards

1
Q

Area Between Curves Formula

A
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2
Q

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

A

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

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3
Q

Find the area of the shaded region

A

343/6

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4
Q

How do you calculate area between curves in terms of y

A

Integral of Right function - Left function

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5
Q

How do you calculate area between curves in terms of x

A

Integral of Top function - Bottom function

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6
Q

If the curves intersect at multiple points…

A

Split the equation into multiple integrals and add them together

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7
Q

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

A

6*sqrt3

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8
Q

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

A

8/3

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9
Q

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

A

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

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10
Q

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

A

22/15

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11
Q

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

A

96pi/5

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12
Q

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

A

2pi/15

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13
Q

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

A

8pi/15

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14
Q

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

A

1/3(l^2*h)

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15
Q

Volume of shell formula

A
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16
Q

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

A

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

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17
Q

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

A

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

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18
Q

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

A

pi/6

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19
Q

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

A

pi/2

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20
Q

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

A

pi/2

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21
Q

Calculus definition of work

A
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22
Q

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.

A

1,000 ft. lbs. or 1355.8 joules

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23
Q

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?

A

1.565 joules

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24
Q

Limit definition of average value of a function

A
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25
Q

Average value integral formula

A
26
Q

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

A

2

27
Q

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

A

Mean value theorem for integrals

28
Q

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

A

c = -1, 1

29
Q

Arc length formula

A
30
Q

Surface area formula

A
31
Q

Function that consists of only natural numbers

A

Sequence

32
Q

Sequence formula

A

sn = s1 + d(n - 1)

n = next term
d = coefficient applied to each term

33
Q

Sum of the elements of a sequence

A

Series

34
Q

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

A

Convergent, convergent, divergent, convergent

35
Q

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

A

Diverges

36
Q

Sum of a geometric series

A

S = a/(1 - r)

37
Q

Geometric series formula

A
38
Q

A series converges if…

A

The absolute value of r is less than 1

39
Q

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

A

Telescoping series

40
Q

Nth partial sum formula

A
41
Q

A series whose terms alternate between positive and negative

A

Alternating series

42
Q

What are the 2 conditions of the alternating series test?

A
43
Q

Alternating Series Estimation Theorem

A
44
Q

Comparison Test Conditions

A
45
Q
A

Based on the comparison test, the series converges

46
Q

Root Test Conditions

A

NOTE: Not commonly used

47
Q

Ratio test conditions

A
48
Q

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

A

Absolutely convergent

49
Q

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

A

Conditionally convergent

50
Q

What test is used given these conditions?

A

Divergence test

51
Q

What test is used given these conditions?

A

Geometric series test

52
Q

What test is used given these conditions?

A

P-series test

53
Q

What test is used given these conditions?

A

Direct comparison test

54
Q

What test is used given these conditions?

A

Limit comparison test

55
Q

What test is used given these conditions?

A

Alternating series test

56
Q

What test is used given these conditions?

A

Alternating series test

57
Q

What test is used given these conditions?

A

Root test

58
Q

What test is used given these conditions?

A

Ratio test

59
Q

What test is used given these conditions?

A

Telescoping series test

60
Q

What test is used given these conditions?

A

Integral test

Nicholas (6 decks)

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