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#?

Answer 1

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

We have:

# y = cscx #

We want to estimate #int \ y \ dx# over the interval #[pi/8,(7pi)/8]# with #3# strips; thus:

# 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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Answer 2

Approximately #3.75# square units

Dividing the given interval #[pi/8,(7pi)/8]#
into 3 equal width intervals (each with a width of #pi/4#:
#color(white)("XXX")I_1=[pi/8,(3pi)/8]#
#color(white)("XXX")I_2=[(3pi)/8,(5pi)/8]#
#color(white)("XXX")i_3=[(5pi)/8,(7pi)/8]#

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

Evaluating #csc(x)# at each interval edge:
#color(white)("XXX")csc(pi/8)=2.6131259298#
#color(white)("XXX")csc((3pi)/8)=1.0823922003 #
#color(white)("XXX")csc((5pi)/8)=1.0823922003 #
#color(white)("XXX")csc((7pi)/8)=2.6131259298#

The area of each interval trapezoid is calculated as
#color(white)("XXX")#the average of the 2 edge lengths times the strip width.

#"Area"_(I1)=(2.6131259298+1.0823922003)/2 xx pi/4#
#color(white)("XXXX")=1.4512265761#

Similarly we can calculate
#"Area"_(I2)=0.8501088462 #

#"Area"_(I3)=1.4512265761#

and the Sum of these Areas gives an approximation of the integral value:
#int_(pi/8)^((7pi)/8)csc(x) dx~~3.7525619983#

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Answer 3

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:

  1. Divide the interval (\left[\frac{\pi}{8}, \frac{7\pi}{8}\right]) into three equal subintervals.
  2. 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.
  3. Evaluate the function (y = \csc(x)) at the endpoints of each subinterval.
  4. 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.
  5. 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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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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