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.

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

step-by-step explanation:

given   that triangle lno is congruent to triangle lnm.

we have to fill up the blanks in the given question.

according to above order of congruency   kept in mind we can say that

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

ln ≅ ln due to common in both.

hence, by sas congruency,

the side angle side rule   states that if two sides and the included angle of one triangle are congruent to another triangle, then these two triangles are congruent.

circumference divided by 3.141592654 then timesthe radius twice

step-by-step explanation:

circumference divided by 3.141592654 then timesthe radius twice