# Estimate the area under the curve #1/(x-1)^2# over the interval #[2,3]# with #n=4# using the trapezium rule?

Trapezium rule gives:

# int_2^3 \ 1/(x-1)^2 \ dx ~~ 0.51 # (2dp)

The values of

Using the trapezoidal rule:

# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n-1))}#

We have:

# int_2^3 \ 1/(x-1)^2 \ dx #

# " " ~~ 0.25/2 { (1 + 0.25) + 2(0.64 + 0.444444 + 0.32653) } #

# " " = 0.125 { 1.25 + 2(1.410975) } #

# " " = 0.125 { 1.25 + 2.82195 } #

# " " = 0.125 { 4.07195 } #

# " " = 0.508993 #

Let's compare this to the exact value:

# int_2^3 \ 1/(x-1)^2 \ dx = [-1/(x-1)]_2^3 #

# " " = -(1/2-1) #

# " " = 0.5 #

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To estimate the area under the curve ( \frac{1}{{(x-1)}^2} ) over the interval ([2,3]) with (n=4) using the trapezium rule, you would first divide the interval ([2,3]) into (n) subintervals of equal width. In this case, with (n=4), each subinterval would have a width of ( \frac{3-2}{4} = 0.25).

Then, apply the trapezium rule formula for each subinterval:

[ A_i = \frac{h}{2} \left( f(x_i) + f(x_{i+1}) \right) ]

where (A_i) is the area of the (i)th trapezium, (h) is the width of each subinterval, and (f(x_i)) and (f(x_{i+1})) are the function values at the endpoints of the (i)th subinterval.

For (i = 1, 2, 3, 4), calculate (f(x_i)) and (f(x_{i+1})) and substitute them into the formula. Finally, sum up the areas of all the trapeziums to get the estimated area under the curve over the interval ([2,3]).

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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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