AP Calculus: Integration flashcards

Question 1

In an integral, what is the integrand?

Accepted Answer

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

Question 2

What do the limits of integration define?

Accepted Answer

The interval over which the function is integrated

Question 3

What is a partition of an interval?

Accepted Answer

A set of points that splits the interval into subintervals

Question 4

What value of f is used on each subinterval for a lower sum?

Accepted Answer

The minimum value of f on that subinterval

Question 5

What value of f is used on each subinterval for an upper sum?

Accepted Answer

The maximum value of f on that subinterval

Question 6

What is the general form of a Riemann sum?

Accepted Answer

The sum from i = 1 to n of f(x_i*) Δx

Question 7

Dividing [a, b] into n equal parts, what is the width Δx?

Accepted Answer

Δx = (b − a)/n

Question 8

What is net signed area?

Accepted Answer

Area above the x-axis minus area below the x-axis

Question 9

What is the sum 1 + 2 + ... + n?

Accepted Answer

n(n + 1)/2

Question 10

What is the sum 1^2 + 2^2 + ... + n^2?

Accepted Answer

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

Question 11

What is the integral of f from a to a?

Accepted Answer

0

Question 12

How does swapping the limits of a definite integral change its value?

Accepted Answer

It changes the sign: the integral from b to a is minus the integral from a to b

Question 13

Are continuous functions on a closed interval integrable?

Accepted Answer

Yes

Question 14

How is the average value of f on [a, b] computed?

Accepted Answer

(1/(b − a)) times the integral of f from a to b

Question 15

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)) times the integral of f from a to b

Question 16

What does Part 1 of the Fundamental Theorem of Calculus say?

Accepted Answer

If f is continuous on [a, b] and F(x) is the integral of f(t) from a to x, then F′(x) = f(x)

Question 17

What does Part 2 of the Fundamental Theorem of Calculus say?

Accepted Answer

If f is continuous on [a, b] and F is any antiderivative of f, the integral of f from a to b is F(b) − F(a)

Question 18

State the net change theorem.

Accepted Answer

F(b) = F(a) + the integral of F′(x) from a to b

Question 19

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

Accepted Answer

0

Question 20

What is ∫ e^x dx?

Accepted Answer

e^x + C

Question 21

What is ∫ a^x dx?

Accepted Answer

a^x/ln a + C

Question 22

What is ∫ x^−1 dx?

Accepted Answer

ln|x| + C

Question 23

What is ∫ ln x dx?

Accepted Answer

x ln x − x + C

Question 24

What is the antiderivative ∫ du/√(a^2 − u^2)?

Accepted Answer

sin^−1(u/a) + C

Question 25

What is ∫ du/(a^2 + u^2)?

Accepted Answer

(1/a) tan^−1(u/a) + C

Question 26

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

Accepted Answer

g(a) and g(b)

AP Calculus: Integration flashcards

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