Definite Integrals

Question 1

What are definite integrals used for?

Answer 1

To find the area under a curve.

Question 2

Riemann Sums

Answer 2

Question 3

Area using left-endpoint rectangles

Answer 3

(b – a)/n • [y₀ + y₁ + y₂ + y₃… + y_n-1]

Question 4

Area using right-endpoint rectangles

Answer 4

(b – a)/n • [y₁ + y₂ + y₃… + y_n]

Question 5

Area using midpoint rectangles

Answer 5

(b-a)/n • [y_1/2 + y_3/2 + y_5/2…+ y_(2n-1)/2]

Question 6

The trapezoidal rule

Answer 6

(1/2) (b – a)/n • [y₀ + 2y₁ + 2y₂ … + 2y_n-1 + y_n]

Question 7

The fundamental theorem of calculus

Answer 7

∫_a^b ƒ(x)dx = F(b) – F(a)

where F(x) is the antiderivative of ƒ(x).

Question 8

The mean value theorem for integrals

Answer 8

If ƒ(x) is continous on a closed interval [a, b], then at some point c in the interval [a, b]:

∫_a^b ƒ(x)dx = ƒ(c)(b – a).

Question 9

Formula for finding the average value of ƒ(x) on [a, b]

Answer 9

ƒ(c) = ¹/_b-a• ∫_a^b ƒ(x)dx.

Question 10

The second fundamental theorem of calculus

Answer 10

If ƒ(x) is continuous on [a, b], then the derviative of the function F(x) = ∫_a^x ƒ(t)dt is:

dF/dx = d/dx ∫_a^x ƒ(t)dt = ƒ(x) dx/dx.