Mathematics, 20.07.2019 20:30 viktoria1198zz

Worth 99 points! use the midpoint rule with n=4 to approximate the area of the region bounded by y=x^3 and y=x. must use the midpoint rule! , you.

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Answer from: wrms8379

The objective is to find total bounded area. Intersections occur at x=-1, x=0, and x=1. Since we need 4 subintervals, we split the line segment from -1 to 1 into four line segment.

That would be $\Delta x = \dfrac{b-a}{n} = \dfrac{1 - \text-1}{4} = \dfrac12$

So subintervals would be
[-1, -0.5]
[-0.5, 0]
[0, 0.5]
[0.5, 1]

Since segment was split evenly, because of rotate symmetry, sub-area for [-1,0] and [0,1] would be same so we can just focus on interval [0,1] with n=2 and then multiple the result by 2.

Now we are talking about midpoint rules, the height of rectangle for [0,0.5], [0.5,1] would be $f(\frac{0+0.5}2)=f(0.25), f(\frac{0.5+1}2)=f(0.75)$ respectively. Width of rectangle would be $\Delta x = 0.5$ .

Hence for

, that would be
$f(0.25)\Delta x + f(0.75)\Delta x = 0.25\cdot0.5 + 0.75\cdot0.5 = 0.5$

And x^3

would be
$(0.25)^3\cdot0.5+(0.75)^3\cdot0.5 = 0.21875$

So the approximate area is 0.28125.

Now double that to account for area from [-1,0] and our answer is 0.5625.

Is this clear?

Answer from: Quest

5. y=6; de=7; fg=8

6. x=4.2; jl=14.2; mn=13

step-by-step explanation:

when chords cross inside a circle, the product of the segment lengths is a constant.

5. dh×eh = fh×gh

y = gh = dh×eh/fh = 3×4/2 . . . divide by the coefficient of gh

y = 6

of course, the total chord length is the sum of the lengths of its segments.

de = dh +eh = 3+4 = 7

fg = fh +gh = 2+6 = 8

6. this problem works the same way as the previous one.

x = lk = mk×nk/jk = 6×7/10 = 4.2

jl = jk +lk = 10 +4.2 = 14.2

mn = mk +nk = 6 +7 = 13

comment on chords and secants

you can think of the points h (problem 5) and k (problem 6) as points where the chords meet. a chord is part of a secant line, a line that intersects the circle in two points. one can also consider these points (h or k) to be the points where the respective secant lines meet.

then, the product that is a constant is the product of the distance from the meeting point (h or k) to one circle intersection with the distance from that meeting point to the other circle intersection. for problem 5, the constant is the product hd×he = hf×hg.

it turns out that this rule regarding the products of lengths to points of intersection is also true when the secants intersect outside the circle.

that is, you only need to remember one rule to work both kinds of problems.

Find the value of the variable and the length of each chord for #5 and #6.

Answer from: Quest

Here, feel free to ask if there are any questions

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Worth 99 points! use the midpoint rule with n=4 to approximate the area of the region bounded by y=...

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