Question

How do you do the midpoint Riemann sum part to find the length of the curve I believe I have the first two parts correct.

Answer 1

`\int_a^b(sinx-xcosx)dx`
`\begin{gathered} a=0 \\ b=\pi \end{gathered}`
`Length\text{ of the curve}\approx3.9981`

Step-by-step explanation:

Given the function;

`\begin{gathered} y=sinx-xcosx\text{ from x = 0 to x = }\pi \\ \end{gathered}`

So to find the area of the curve defined by the above, we have to compute;

`\int_a^b(sinx-xcosx)\text{dx where a = 0 and b = }\pi`

Let's go ahead and determine the length of the curve as seen below;

`\begin{gathered} \int_0^(\pi)(sinx-xcosx)dx \\ =\int_0^(\pi)sinxdx-\int_0^(\pi)xcosxdx=[-cosx]_0^(\pi)-[xsinx+cosx]_0^(\pi) \\ =[-cos(\pi)-(-cos(0))]-[(\pi sin\pi+cos\pi)-(0sin0+cos0)] \\ =[-(-1)-(-1)]-[(\pi(0)+(-1))-[0+(1)] \\ =2-(-2) \\ =2+2 \\ =4 \end{gathered}`

Given the below midpoint Riemann sum formula;

`A_m=\sum_{i\mathop{=}1}^n\Delta xf(x_i)`

where;

`\begin{gathered} A=area\text{ under the curve} \\ \Delta x=width\text{ of each each rectangle} \\ f(x)=length\text{ of each rectangle} \\ n=number\text{ of rectangles = 4} \end{gathered}`

Let's determine the width of each rectangle as seen below;

`\begin{gathered} \Delta x=(b-a)/(n)=(\pi-0)/(4)=(\pi)/(4) \\ So\text{ the middle points will }(\pi)/(8),(3\pi)/(8),(5\pi)/(8),(7\pi)/(8) \end{gathered}`

We can now go ahead and solve for A as seen below;

`\begin{gathered} A_m=\Delta x[f((\pi)/(8))+f((3\pi)/(8))+f((5\pi)/(8))+f((7\pi)/(8))] \\ A_m=(\pi)/(4)(0.0199+0.4730+1.6753+2.9223) \\ A_m=0.7854(5.0905) \\ A_m=3.9981 \end{gathered}`

So an approximate value of the length of the curve using the middle Riemann sum is 3.9981