Use simpson's rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. compare your answer with the value of the integral produced by a calculator. (round your answers to six decimal places.) y = 5xex, 0 ≤ x ≤ 1
a. simpson's rule
b. calculator approximation

= (1/5) x^5 => dy/dx = x^4

dS = dx * sqrt[1 + (dy/dx)^2]

= dx * sqrt(1 + x^4).

So the exact surface area is the integral from x=0 to x=5 of

sqrt(1+x^4) dx.

Since this cannot be integrated analytically in terms of "elementary" functions (it comes out including an elliptic integral, which is a tabulated function) -- the treatment by a numerical method makes sense.

Simpson's Rule dictates that each subinterval [a,b] be treated by approximating the area there as

[(b-a)/6] * [f(a) + 4f(0.5a + 0.5b) + f(b)]

In your problem, it's not obvious whether the subintervals should be chosen as equal intervals of length 1/2, or whether they should be crowded towards the left end of [0,5], since the curvature is concentrated at the left end. What militates against the latter choice is that the area is so much larger towards the right end. Since you're answering a "school" question, I'll play it safe and choose 1/2 as the uniform interval-length.

[0,1/2] area = (1/12) [1 + 4*sqrt(1 + 1/256) + sqrt(1+1/16)]

= (1/12)*6.03858 = 0.50643

[1/2,1] area = (1/12) [sqrt(1+1/16) + 4*sqrt(1 + 81/256) + sqrt(2)]

= (1/12)*7.03438 = 0.58640

[1,3/2] area = (1/12) [sqrt(2) + 4*sqrt(1+625/256) + sqrt(1+81/16)]

= (1/12)*11.29684 = 0.94160

OK, those are the first three (of 10) subintervals. You can do the other 7, and then add all ten results together.

For a check on reasonableness, I used fooplot to graph y=sqrt(1+x^4) from (0,1) to (5,25.02), and then also plotted y=(24/5)+1 from (0,1) to (5,25). The area under the latter is of course 5*13 = 65, and it can be seen from the plot that the area under sqrt(1+x^4) is substantially smaller, but not ENORMOUSLY smaller; it might be somewhere around 50.

hey there!

if she's saving for six months, let's divide the total number we need for the trip by six, dividing it into the amount that needs to be saved for each month.

500 ÷ 6 =

she needs to save at least this amount each month, so we should use this symbol: ≥

let x be the amount she saves each month.

your final answer is x ≥

hope this !

1/800 is the right answer