Estimate the area under #y=x^2+x# from #x=0.2# to #x=1# using Simpson's rule with #6# strips?
# A = int_0.2^1 \ x^2+x \ dx ~~ 0.810667 #
We have:
# y = x^2+x #
We want to estimate
# Deltax = (10.2)/6 = 2/15#
The values of the function are tabulated as follows;
Simpson's Rule
Using Simpson's Rule we can estimate the area using best fit quadratics, which generally gives an accurate estimation. We use the formula:
# int_a^b \ f(x) \ dx ~~ h/3{(y_0+y_n)+4(y_1+y_3+...+y_(n1))+2(y_2+Y_4+...+y_(n2))}#
So we have:
# A = int_0.2^1 \ x^2+x \ dx #
# \ \ \ ~~ 0.133333/3 { 0.24 + 2 + 4(0.444444 + 0.96 + 1.617778)+2(0.684444 + 1.271111)} #
# \ \ \ = 0.044444 { 2.24 + 4(3.022222)+ 2(1.955556)} #
# \ \ \ = 0.044444 { 2.24 + 12.088889+3.911111 } #
# \ \ \ = 0.066667 * 18.24 #
# \ \ \ = 0.810667 #
Actual Value
For comparison of accuracy:
# A = int_0.2^1 \ x^2+x \ dx #
# \ \ \ = [x^3/3+x^2/2]_0.2^1 #
# \ \ \ = (1/3+1/2) (1/375+1/50)#
# \ \ \ = 304/375 #
# \ \ \ = 0.810666666...#
Notes
We note that to the accuracy of calculation Simpsons rules produced an exact answer. This is no conicenice. The function we are integrating is quadratic, and Simpsons rils works by estimating the area under the best fit quadratic of the adjacent points and therefore it provided, in this case, the exact answer.
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To estimate the area under the curve (y = x^2 + x) from (x = 0.2) to (x = 1) using Simpson's rule with 6 strips, you can follow these steps:

Determine the interval width, (h), which is given by: [h = \frac{{b  a}}{n}] where (a) and (b) are the limits of integration (in this case, (a = 0.2) and (b = 1)), and (n) is the number of strips (given as 6). So, (h = \frac{{1  0.2}}{6} = 0.1333).

Calculate the function values at each interval boundary (including (a) and (b)), (x_i), using (y = x^2 + x).

Use Simpson's rule formula to calculate the area: [A \approx \frac{h}{3}[y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6]] where (y_0) is the function value at (x = a), (y_1, y_3, y_5) are the function values at the odd indices, and (y_2, y_4, y_6) are the function values at the even indices.

Plug in the function values into the formula and calculate the approximate area.

Simplify the expression to find the numerical value of the area.
Following these steps, you can estimate the area under the curve (y = x^2 + x) from (x = 0.2) to (x = 1) using Simpson's rule with 6 strips.
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To estimate the area under the curve ( y = x^2 + x ) from ( x = 0.2 ) to ( x = 1 ) using Simpson's rule with 6 strips, follow these steps:

Divide the interval from ( x = 0.2 ) to ( x = 1 ) into 6 equal strips. The width of each strip (( h )) will be ( \frac{1  0.2}{6} = 0.1333 ).

Calculate the function values ( y_i ) at each of the strip boundaries using the given function ( y = x^2 + x ). For simplicity, you can choose either the left or right endpoint of each strip.

Apply Simpson's rule formula to each pair of strips to estimate the area. Simpson's rule formula for two adjacent strips is:
[ A_i = \frac{h}{3} \left( y_i + 4y_{i+\frac{1}{2}} + y_{i+1} \right) ]
 Sum up the areas obtained from each pair of strips to get the final estimate for the total area under the curve.
By following these steps, you can accurately estimate the area under the curve using Simpson's rule with 6 strips.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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