Consider the function $f (x) = - {(x - 3)}^{2} + 4$ $f(x)= -(x-3)^2+4$ how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?

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Consider the function $f (x) = - {(x - 3)}^{2} + 4$ $f(x)= -(x-3)^2+4$ how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?

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Answer 1 · 2017-08-14T19:17:34

$A = lim_(nrarroo)sum_(i=1)^n (-(i4/n-2)^2+4) 4/n$

Explanation:

The curve intersects the $x$ axis at $f(x) =0$

or $x=1$ and $x=5$ .

Cut the interval $[1,5]$ into $n$ pieces each of length $(5-1)/n = 4/n$

The right endpoints of the subintervals are $1+(4i)/n$ .

We can approximate the area under the curve on the $i^(th)$ interval using a rectangle of

base $4/n$ and

height $f(1+(4i)/n) = -(1+(4i)/n-3)^2+4 = -(i4/n-2)^2+4$ .

We then sum the areas of the rectangles $sum_(i=1)^n (-(i4/n-2)^2+4) 4/n$ .

Finally, we take a limit as the subintervals get shorter and shorter (go to $0$ ). This is aso the limit as the number of rectangles increases without bound ( $nrarroo$ )

$A = lim_(nrarroo)sum_(i=1)^n (-(i4/n-2)^2+4) 4/n$

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Consider the function $f (x) = - {(x - 3)}^{2} + 4$ $f(x)= -(x-3)^2+4$ how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?

1 Answer

Explanation:

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Math

Humanities

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Consider the function f(x)= -(x-3)^2+4f(x)=−(x−3)2+4f(x)= -(x-3)^2+4 how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?

1 Answer

Explanation:

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Consider the function $f (x) = - {(x - 3)}^{2} + 4$ $f(x)= -(x-3)^2+4$ how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?