Calculus: Integration flashcards

Question 1

Over [a,b] split into n equal parts, what is the width Δx of each rectangle?

Accepted Answer

Δx = (b − a)/n

Question 2

What is a partition of an interval?

Accepted Answer

A set of points dividing the interval into subintervals

Question 3

How is the definite integral of f from a to b defined as a limit?

Accepted Answer

lim as n→∞ of Σ f(x_i*)Δx, for i = 1 to n

Question 4

What is the integrand of an integral?

Accepted Answer

The function to the right of the integral sign, the one being integrated

Question 5

What do the limits of integration specify?

Accepted Answer

The interval over which the function is integrated

Question 6

What is net signed area?

Accepted Answer

The area above the x-axis minus the area below it; it equals the definite integral

Question 7

Are continuous functions on a closed interval integrable?

Accepted Answer

Yes

Question 8

What is the integral of f(x) from a to a?

Accepted Answer

0

Question 9

How does swapping the limits a and b change a definite integral?

Accepted Answer

It changes its sign: ∫ from b to a = −∫ from a to b

Question 10

How can ∫ from a to b of f(x) dx be split at a point c?

Accepted Answer

∫ from a to c of f(x) dx + ∫ from c to b of f(x) dx

Question 11

State the Mean Value Theorem for Integrals.

Accepted Answer

If f is continuous on [a,b], some c in [a,b] has f(c) = (1/(b − a))∫ from a to b of f(x) dx.

Question 12

By the Fundamental Theorem of Calculus, Part 1, what is F′(x) if F(x) = ∫ from a to x of f(t) dt with f continuous on [a,b]?

Accepted Answer

F′(x) = f(x)

Question 13

By the Fundamental Theorem of Calculus, Part 2, what is ∫ from a to b of f(x) dx for f continuous on [a,b] with antiderivative F?

Accepted Answer

F(b) − F(a)

Question 14

How does the net change theorem write F(b)?

Accepted Answer

F(b) = F(a) + ∫ from a to b of F′(x) dx

Question 15

What is the integral of an odd function over a symmetric interval?

Accepted Answer

Zero

Question 16

What is the sum 1 + 2 + … + n?

Accepted Answer

n(n + 1)/2

Question 17

What is the sum 1^2 + 2^2 + … + n^2?

Accepted Answer

n(n + 1)(2n + 1)/6

Question 18

What is ∫ e^x dx?

Accepted Answer

e^x + C

Question 19

What is ∫ a^x dx?

Accepted Answer

a^x/ln a + C

Question 20

What is ∫ x^−1 dx?

Accepted Answer

ln|x| + C

Question 21

What is ∫ ln x dx?

Accepted Answer

x ln x − x + C

Question 22

What is ∫ du/√(a^2 − u^2)?

Accepted Answer

sin^−1(u/a) + C

Question 23

What is ∫ du/(a^2 + u^2)?

Accepted Answer

(1/a) tan^−1(u/a) + C

Question 24

With u = g(x) in a definite integral from a to b, what become the new limits?

Accepted Answer

g(a) and g(b)

Calculus: Integration flashcards

The cards

Where this deck sits in the curriculum

How this deck was checked

Sources

More in this course