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?
Finally,
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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:
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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 ]
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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.
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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.} ]
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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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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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- How do you find the error that occurs when the area between the curve #y=x^3+1# and the x-axis over the interval [0,1] is approximated by the trapezoid rule with n = 4?
- If #f(x)=x^(1/2)#, #1 <= x <= 4# approximate the area under the curve using ten approximating rectangles of equal widths and left endpoints?

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