# How do you use the Trapezoidal rule and three subintervals to give an estimate for the area between #y=cscx# and the x-axis from #x= pi/8# to #x = 7pi/8#?

# int_(pi/8)^((7pi)/8) \ cscx \ dx ~~ 3.7526 \ \ (4dp)#

We have:

# y = cscx #

We want to estimate

# Deltax = ((7pi)/8-pi/8)/3 = pi/4#

The values of the function, working to 6dp, are tabulated using Excel as follows;

Trapezium Rule

# A = int_(pi/8)^((7pi)/8) \ cscx \ dx #

# \ \ \ ~~ 0.785398/2 * { 2.613126 + 2.613126 + 2*(1.082392 + 1.082392) } #

# \ \ \ = 0.392699 * { 5.226252 + 2*(2.164784) } #

# \ \ \ = 0.392699 * { 5.226252 + 4.329569 } #

# \ \ \ = 0.392699 * 9.555821 #

# \ \ \ = 3.752562 #

Actual Value

For comparison of accuracy:

# A= int_(pi/8)^((7pi)/8) \ cscx \ dx #

# \ \ \ = [ color(white)(""/"") -log(abs(csc(x)+cot(x))) \ ]_(pi/8)^((7pi)/8 #

# \ \ \ = 3.229781832346191 #

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Approximately

Dividing the given interval

into 3 equal width intervals (each with a width of

[from this point on, extensive use of spreadsheet/calculator is recommended]

Evaluating

The area of each interval trapezoid is calculated as

Similarly we can calculate

and the Sum of these Areas gives an approximation of the integral value:

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To use the Trapezoidal rule with three subintervals to estimate the area between (y = \csc(x)) and the x-axis from (x = \frac{\pi}{8}) to (x = \frac{7\pi}{8}), follow these steps:

- Divide the interval (\left[\frac{\pi}{8}, \frac{7\pi}{8}\right]) into three equal subintervals.
- Calculate the width of each subinterval ((h)) using the formula: (h = \frac{b-a}{n}), where (b) is the upper limit of integration, (a) is the lower limit of integration, and (n) is the number of subintervals.
- Evaluate the function (y = \csc(x)) at the endpoints of each subinterval.
- Use the Trapezoidal rule formula for each subinterval: [A_i = \frac{h}{2}(y_i + y_{i+1})] where (A_i) is the area of the (i)-th trapezoid, (h) is the width of the subinterval, (y_i) is the function value at the lower endpoint, and (y_{i+1}) is the function value at the upper endpoint.
- Sum up the areas of the three trapezoids to get the estimate for the total area.

Following these steps, you'll get an estimate for the area between (y = \csc(x)) and the x-axis from (x = \frac{\pi}{8}) to (x = \frac{7\pi}{8}).

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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.

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