Question

can anyone help me with this please?



Write the Riemann sum to find the area under the graph of the function f(x) = x4 from x = 5 to x = 7.

Answer 1

\(x^4\) ?

Answer 2

\(\color{#000000 }{ \displaystyle \int\limits_5^7x^4~dx }\)

That is basically your question, and you would like to do this geometrically (right?). Are you given the method to use, and the number of shapes you must use, or anything?

Answer 3

that's all the question gives me

Answer 4

So, you can do the right end, left end, sum, OR, upper sum, lower sum?? whichever one you choose?

Answer 5

Well, I think that we can use n=4.

\(\color{#000000 }{ \displaystyle \Delta x=\frac{b-a}{n}=\frac{7-5}{4}=\frac{1}{2} }\)

Answer 6

these are the choices

Answer 7

Oh, there are options. This is different:)

Answer 8

Yes, should've added it with the question :P

Answer 9

\(\color{#000000 }{ \displaystyle \int\limits_a^b f(x){\tiny~}{\mathrm d}x= \lim_{n\to\infty}\left[\sum_{i=0}^{n}f\left(x_i\right)\Delta x_i\right]}\)

that is the general form; I think I got that right:)

Answer 10

\(\color{#000000 }{ \displaystyle \Delta x_i=\frac{a-b}{n}=\frac{7-5}{n}=\frac{2}{n}}\)

\(\color{#000000 }{ \displaystyle c_i=b+\left(\frac{b-a}{n}\right)i=7+\left(\frac{7-5}{n}\right)i=\frac{2i}{n}}\)

Answer 11

oh, that \(x_i\), I wrote it as \(c_i\).... I apologize for that. sorry

Answer 12

ok, I understand that part thank you

Answer 13

\(\color{#000000 }{ \displaystyle y(\omega) }\)

Answer 14

Actually, to be precise and simple, you simply get:

\(\color{#000000 }{ \displaystyle \int\limits_5^7{\tiny~}x^4{\tiny~}dx=\left.\frac{x^{4\color{red}{+1}}}{4\color{red}{+1}}\right|_{x{\tiny~~}={\tiny~~}5}^{x{\tiny~~}={\tiny~~}7} =\left.\frac{x^5}{5}\right|_{x{\tiny~~}={\tiny~~}\color{blue}{5}}^{x{\tiny~~}={\tiny~~}{\color{blue}{7}}} =\frac{(\color{blue}{7})^5}{5}-\frac{(\color{blue}{5})^5}{5}=2736.4}\)

Answer 15

that is all you would normally need to do to find this area,
using the \(\color{blue}{\rm power~~rule~~of~~integration}\), and by applying
\(\color{blue}{\rm fundamental~~theorem~~of~~calculus}\).

Answer 16

thank you so much, I was so lost!

Answer 17

\(\color{#000000 }{ \displaystyle \int\limits_a^b{\tiny~}x^n{\tiny~}dx=\left.\frac{x^{n\color{red}{+1}}}{n\color{red}{+1}}\right|_{x{\tiny~}={\tiny~}a}^{x{\tiny~}={\tiny~}b}=\frac{b^{n\color{red}{+1}}}{n\color{red}{+1}}-\frac{a^{n\color{red}{+1}}}{n\color{red}{+1}}}\)

that is the power rule, by a definite integral.

Answer 18

would the answer be d?

Answer 19

yes, for this purpose, yes.

Answer 20

And the fundamental theorem of calculus

\(\color{#000000 }{ \displaystyle \left. \int\limits_a^b{\tiny~}f'(x) {\tiny~}dx=f(x)\right|_{x~=~a}^{x~=~b}=f(b)-f(a)}\)

Answer 21

I understand, thank you! could I possibly ask you for help with one more?

Answer 22

I will try, if I won't have to leave:)

Answer 23

thank you!

Answer 24

\(\color{#000000 }{ \displaystyle \int\limits_0^{10}f(x)~dx }\)

\(\color{#000000}{f(x)=\begin{cases} 5,\quad\quad\quad~ x\le 5 \\ 10-x,\quad x >5 \end{cases} }\)

Answer 25

Can you please re-write this integral, as the sum of two functions, based on the piece wise function that you are given?

Answer 26

(I mean as the sum of two integrals of two functions)

Answer 27

wouldn't they both equal 50?

Answer 28

I don't know yet ... can you show me how you got this result please?

Answer 29

I did 5(10)-0