How do you use the trapezoidal rule with n=60 to approximate the area between the curve #y=sinx# from 0 to pi?

Answer 1

See the explanation.

There is no "area between a curve". I assume you want the area beneath the curve (and above the #x# axis).

We use the trapezoidal rule by using the formula.

#T_60 = 1/2 Deltax(f(x_0)+2f(x_1)+2f(x_3)+* * * +2f(x_(n-1)+f(x_n))#
In this case #f(x) = sinx#
#n=60# and #Deltax = (b-a)/n = (pi-0)/60 = pi/60#
#x_0 = 0, x_1=pi/60, x_2 = (2pi)/60, x_3=(3pi)/60, . . . x_(n-1)=((n-1)pi)/60, x_n = pi#
Plug in the numbers and do the arithmetic. (If permitted, use a computer spreadsheet for all that arithmetic. I got #1.999543#, which is quite close to the exact value of #2#)
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Answer 2

Using the trapezoidal rule with n=60 to approximate the area between the curve y=sinx from 0 to π involves dividing the interval [0, π] into 60 equal subintervals, each with a width of Δx = (π - 0) / 60. Then, calculate the height of each trapezoid by evaluating sin(x) at the endpoints of each subinterval. Finally, sum up the areas of all the trapezoids to get the approximation of the desired 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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