Homework: HW 5 Score: 0 of 1 pt 7.6.11 Save 28 of 28 (25 complete) HW Score: 83.

Question

Homework: HW 5 Score: 0 of 1 pt 7.6.11 Save 28 of 28 (25 complete) HW Score: 83.04%, 23.25 of 28 pts Question Help Find the area under the semicircle y =-/36-x2 and above the x-axis by using n-8 by the following methods: (a) the trapezoidal rule, and (b) Simpson's rule. (c) Compare the results with the area found by the formula for the area of a circle. (a) Use the trapezoidal rule to approximate the area under the semicircle (Round the final answer to three decimal places as needed. Round all intermediate values to four decimal places as needed.)

Explanation / Answer

a)

Here a= -6, b= 6

?x = (6 - -6)/8
?x = 12/8
?x = 1.5

Hence

Trapezoidal approximation =(1/2) * (1.5) * ( f(-6) + 2*f(-4.5) + 2*f(-3) + 2*f(-1.5) + 2*f(0) + 2*f(1.5) + 2*f(3) + 2*f(4.5) + f(6) )

=(1/2) * (1.5) * ( 0 + 2*3.9686269665969 + 2*5.1961524227066 + 2*5.8094750193111 + 2*6 + 2*5.8094750193111 + 2*5.1961524227066 + 2*3.9686269665969 + 0 )

=53.923

b)

Simpson =(1/3)* (1.5) * ( f(-6) + 4*f(-4.5) + 2*f(-3) + 4*f(-1.5) + 2*f(0) + 4*f(1.5) + 2*f(3) + 4*f(4.5) + f(6) )

=(1/3)* (1.5) * ( 0 + 4*3.9686269665969 + 2*5.1961524227066 + 4*5.8094750193111 + 2*6 + 4*5.8094750193111 + 2*5.1961524227066 + 4*3.9686269665969 + 0 )

= 55.505

c)

exact area is

A= (1/2) *pi*6² = 18pi = 56.5486

so both method Underestimate the actual area.

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