Fellow brains pls help.... Is this true or false?

Which definite integral approximation formula is this: the integral from a to b of f(x)dx ≈ (b-a)/n * [<img src="https://tex.z-d

Stella [2.4K]

The answer is most likely A.

The integration interval [a, b] is split up into n subintervals of equal length (so each subinterval has width (b - a)/n, same as the coefficient of the sum of y terms) and approximated by the area of n rectangles with base (b - a)/n and height y.

n subintervals require n + 1 points, with

x₀ = a

x₁ = a + (b - a)/n

x₂ = a + 2(b - a)/n

and so on up to the last point x = b. The right endpoints are x₁, x₂, … etc. and the height of each rectangle are the corresponding y 's at these endpoints. Then you get the formula as given in the photo.

• "Average rate of change" isn't really relevant here. The AROC of a function G(x) continuous* over an interval [a, b] is equal to the slope of the secant line through x = a and x = b, i.e. the value of the difference quotient

(G(b) - G(a) ) / (b - a)

If G(x) happens to be the antiderivative of a function g(x), then this is the same as the average value of g(x) on the same interval,

$g_{\rm ave}=\dfrac{G(b)-G(a)}{b-a}=\dfrac1{b-a}\displaystyle\int_a^b g(x)\,\mathrm dx$

(* I'm actually not totally sure that continuity is necessary for the AROC to exist; I've asked this question before without getting a particularly satisfying answer.)

• "Trapezoidal rule" doesn't apply here. Split up [a, b] into n subintervals of equal width (b - a)/n. Over the first subinterval, the area of a trapezoid with "bases" y₀ and y₁ and "height" (b - a)/n is

(y₀ + y₁) (b - a)/n

but y₀ is clearly missing in the sum, and also the next term in the sum would be

(y₁ + y₂) (b - a)/n

the sum of these two areas would reduce to

(b - a)/n = (y₀ + 2 y₁ + y₂)

which would mean all the terms in-between would need to be doubled as well to get

$\displaystyle\int_a^b f(x)\,\mathrm dx\approx\frac{b-a}n\left(y_0+2y_1+2y_2+\cdots+2y_{n-1}+y_n\right)$

7 0

3 years ago

PLS ANSWER WHAT IS 3 and 14 hundredths minus 1 and 9 hundredths?

k0ka [10]

Answer:

2 and 5 hundredths

Step-by-step explanation:

3 - 1 = 2

14 hundredths - 9 hundredths = 5 hundredths

hence you get 2 and 5 hundredths

6 0

3 years ago

Julie is 3 years older than her sister. Julie's age is 11 years old. How old is her sister?

Ne4ueva [31]

Answer:

14 :)

Step-by-step explanation:

4 0

3 years ago

This is Anya's solution for the equation x ^ 2 - 6x - 7 = 0 Is her solution correct ? Explain your answer in detail points ANSWE

tia_tia [17]

Answer:

Anya is incorrect.

Step-by-step explanation:

=)

The correct answer is x = −1 or x = 7

=)

5 0

3 years ago

Read 2 more answers

Terry made 6 orders in September, 36 orders in October, and 216 orders in November. If this pattern continues, how many orders w

bixtya [17]

Answer:

1296 orders

Step-by-step explanation:

There is a pattern to his orders.

First, he starts with 6.

Second, he made 36.

Third, he made 216.

From here, the pattern can be identified. And, you can extend it further.

Pattern - x · 6

x = the orders.

6, 36, 216

6 (September) × 6 = 36 (October)

36 × 6 = 216 (November)

216 × 6 = 1296 (December)

7 0

4 years ago