Consider the function #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?

Answer 1

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

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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Answer 2

To determine the area enclosed by the function f(x) and the x-axis, we can use a definite integral. The integral of f(x) from a to b represents the area enclosed between the curve and the x-axis over the interval [a, b]. In this case, we need to find the limits of integration, which are the x-values where f(x) intersects the x-axis. To find these points, we set f(x) equal to zero and solve for x. In this case, -(x-3)^2+4 = 0. By solving this equation, we can find the x-values where f(x) intersects the x-axis. Once we have these limits of integration, we can evaluate the definite integral to find the area enclosed by f(x) and the x-axis.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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