How do you find the area bounded by #y=4-x^2#, the x and y axis, and x=1?

Answer 1

First picture what this region would look like by envisioning its graph (or just looking straight at it):

graph{4-x^2 [-9.54, 10.46, -3.92, 6.08]}

Thinking about how this is bounded from side to side, we see it's bounded by the #y#-axis and the line #x=1#.
Since it's also bounded by the #x#-axis, we're looking for the positive area in that "slice" of the graph between #0lt=xlt=1#:

graph{(4-x^2)sqrt(x-x^2)/sqrt(x-x^2) [-2, 3, -1, 4.72]}

This are can be found through integrating the function from #x=0# to #x=1#, or:
#int_0^1(4-x^2)dx#

Integrating (finding the antiderivative) and keeping the bounds gives:

#=[4x-x^3/3]_0^1#
#=[4(1)-1^3/3]-[4(0)-0^3/3]#
#=(4-1/3)-(0-0)#
#=11/3#
The area under the specified curve with the specified bounds is #11/3#.
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Answer 2

To find the area bounded by the curve y = 4 - x^2, the x and y axes, and the line x = 1, we integrate the function from the x-axis up to the curve and then subtract the area under the x-axis (which is negative).

The integral setup is: ∫[0 to 1] (4 - x^2) dx.

Evaluating this integral gives the area bounded by the curve, the x-axis, and the line x = 1.

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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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