Question

Integration by parts problem? help?

Consider ∫ x^2 g(x) dx from 0 to 10 , where g has the values in the following table.

x : 0 2 4 6 8 10
g(x) : 2.4 3 4.4 5.6 6.2 6.8

(a) Which of the following represents the first step using integration by parts?

>> I got .. x^2 g(x) (from 0 to 10) - ∫ 2x g(x) dx (from 0 to 10)

(b) Approximate the integral. Use the right hand Riemann sum with n = 5 to approximate the second term in your answer to part (a)?

How to get (b) with steps please :)

Also, there is a picture attached for the problem (clear).
https://s.yimg.com/hd/answers/i/5a950

Answer 1

\[I=\int_0^{10}x^2g(x)~dx\]
Integrating by parts, you'd let
\[\begin{matrix}u=x^2&&&dv=g(x)~dx\\du=2x~dx&&&v=\int g(x)~dx\end{matrix}\]
So we have
\[I=\bigg[x^2\int g(x)~dx\bigg]_0^{10}-\int_0^{10}2x\left(\int g(x)~dx\right)~dx\]
See the problem here?

The fact that the table lists values for \(g(x)\) makes me wonder ... Are you sure the original integral isn't
\[\int_0^{10}x^2g'(x)~dx~~?\]

Answer 2

Yes it is. Please look at the picture in here
https://answers.yahoo.com/question/index?qid=20140521145252AAdHi4Y

Answer 3

\[I=\int_0^{10}x^2g'(x)~dx\]
Integrating by parts, you'd let
\[\begin{matrix}u=x^2&&&dv=g'(x)~dx\\du=2x~dx&&&v=g(x)\end{matrix}\]
So we have
\[I=\bigg[x^2g(x)\bigg]_0^{10}-\int_0^{10}2x~g(x)~dx\]
Right, so your choice for (a) was correct.

Simplifying a bit, we get
\[I=100g(10)-2\int_0^{10}x~g(x)~dx\\
I=680-2\int_0^{10}x~g(x)~dx\]

Answer 4

For the part on approximation, let's suppose the sketch below models \(x~g(x)\):