How do you use the trapezoidal rule with n=3 to approximate the area between the curve y=x^2 and the x-axis for 1 ≤ t ≤ 4?

Answer 1

#"Area"=22.125" square units"#

NB : Assuming that the interval is #color(blue)(1<=x<=4)#and not #color(red)(1<=t<=4)#
The trapezoidal rule says that the area can be found using the formula: #"Area"=h/2(y_1+y_n+2(y_2+y_3+...+y_(n-1)))#
where #h# is the step length(the length of a single strip), #y_1,y_2,...,y_n# are the #y# values corresponding to each #x# value taken from the interval #[1,4]#
In this problem, since #n=3#, #"Area"=h/2(y_1+y_3+2(y_2))#
First, #h=(x_n-x_1)/(n-1)#
#x_1# and #x_n# are respectively the first and last #x# values in the interval
#h=(4-1)/(3-1)=3/2#
Next, #y_1=(x_1)^2=(1)^2=1# #y_2=(1+3/2)^2=(5/2)^2=25/4# #y_3=(5/2+3/2)^2=(4)^2=16#
NB : In order to get the next #x# we add #3/2# to the previous. Example : #x_2=1+3/2=5/2#

Finally,

#"Area"=(3/2)/2(1+16+2(25/4))~=color(blue)(22.125)#
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Answer 2

To use the trapezoidal rule with ( n = 3 ) to approximate the area between the curve ( y = x^2 ) and the x-axis for ( 1 \leq t \leq 4 ), follow these steps:

  1. Divide the interval ( [1, 4] ) into ( n = 3 ) equal subintervals. The width of each subinterval, ( \Delta x ), is calculated as: [ \Delta x = \frac{b - a}{n} = \frac{4 - 1}{3} = \frac{3}{3} = 1 ]

  2. Determine the function values at the endpoints and the midpoints of each subinterval. In this case, evaluate ( y = x^2 ) at ( x = 1, 2, 3, ) and ( 4 ) to get ( y = 1, 4, 9, ) and ( 16 ), respectively.

  3. Apply the trapezoidal rule formula to each pair of consecutive points: [ A_i = \frac{1}{2}(\text{height}i + \text{height}{i+1}) \times \text{base}_i ] [ \text{where } A_i \text{ is the area of the trapezoid for the } i\text{th subinterval.} ]

  4. Sum up the areas of all the trapezoids to get the approximate area under the curve.

[ \text{Approximate area} \approx A_1 + A_2 + A_3 ]

Substitute the values calculated in steps 2 and 3 into the formula to find the approximate area.

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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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