Question

Express the given integral as the limit of a Riemann sum but do not evaluate

Answer 1

That is the integral

Answer 2

all opinions appreciated

Answer 3

\[\int_a^b f(x) dx=\lim_{ n \rightarrow \infty} \sum_{i=1}^{n}\frac{b-a}{n}f(a+i \cdot \frac{b-a}{n})\]

Answer 4

@myininaya
the first one didn't come up

Answer 5

i got it

Answer 6

you should be able to identify a and b and f(x) easily from your integral

Answer 7

can you tell me what they are?

Answer 8

a = 0
b = 3

Answer 9

ok and what is f(x)

Answer 10

f(x) = x^3 - 6x

Answer 11

right we will need to evaluate f(x)=x^3-6x for where x=a+i*(b-a)n

Answer 12

\[f(x)=x^3-6x \\ f(a+i \cdot \frac{b-a}{n})=f(0+i \cdot \frac{3-0}{n})=f(i \frac{3}{n})=?\]

Answer 13

replace the x's with (3i/n)

Answer 14

(3i/n)^3 - 6(3i/n)

Answer 15

right
\[f(x)=x^3-6x \\ f(a+i \cdot \frac{b-a}{n})=(\frac{3i}{n})^3-6(\frac{3i}{n}) \text{ where } a=0, b=3 \\ \text{ you can simplify a little but \not much } \\ f(a+i \cdot \frac{b-a}{n})=\frac{3^3 i^3}{n^3}-\frac{6 \cdot 3 i}{n}\]
you simplify 3^3 and 6*3 and that is all we really need to do besides pluggin this into the equation I have above (the first equation I wrote)

Answer 16

\[\int\limits_a^b f(x) dx=\lim_{ n \rightarrow \infty} \sum_{i=1}^{n}\frac{b-a}{n}f(a+i \cdot \frac{b-a}{n}) \]
so you only need to replace b and a and f(a+i*(b-a)/n) with what we just found for it

Answer 17

don't forgot to put ( ) around the thing we just found for f(a+i*(b-a)/n) since that whole output is to be multiplied by the (b-a)/n thing

Answer 18

Thanks

Answer 19

np