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2 answers:
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From the figure shown, the interval is divided into 5 equal parts making each subinterval to be 0.2.
Part A:
The approximate the area of the region shown in the figure using the lower sums is given by:
![Area= [y(0.2)\times0.2]+[y(0.4)\times0.2]+[y(0.6)\times0.2]+[y(0.8)\times0.2] \\ +[y(1)\times0.2] \\ \\ =[\sqrt{1-(0.2)^2}\times0.2]+[\sqrt{1-(0.4)^2}\times0.2]+[\sqrt{1-(0.6)^2}\times0.2] \\ +[\sqrt{1-(0.8)^2}\times0.2]+[\sqrt{1-(1)^2}\times0.2] \\ \\ =(0.9798\times0.2)+(0.9165\times0.2)+(0.8\times0.2)+(0.6\times0.2)+(0\times0.2) \\ \\ =0.196+0.183+0.16+0.12=0.659](https://tex.z-dn.net/?f=Area%3D%20%5By%280.2%29%5Ctimes0.2%5D%2B%5By%280.4%29%5Ctimes0.2%5D%2B%5By%280.6%29%5Ctimes0.2%5D%2B%5By%280.8%29%5Ctimes0.2%5D%20%5C%5C%20%2B%5By%281%29%5Ctimes0.2%5D%20%5C%5C%20%20%5C%5C%20%3D%5B%5Csqrt%7B1-%280.2%29%5E2%7D%5Ctimes0.2%5D%2B%5B%5Csqrt%7B1-%280.4%29%5E2%7D%5Ctimes0.2%5D%2B%5B%5Csqrt%7B1-%280.6%29%5E2%7D%5Ctimes0.2%5D%20%5C%5C%20%2B%5B%5Csqrt%7B1-%280.8%29%5E2%7D%5Ctimes0.2%5D%2B%5B%5Csqrt%7B1-%281%29%5E2%7D%5Ctimes0.2%5D%20%5C%5C%20%20%5C%5C%20%3D%280.9798%5Ctimes0.2%29%2B%280.9165%5Ctimes0.2%29%2B%280.8%5Ctimes0.2%29%2B%280.6%5Ctimes0.2%29%2B%280%5Ctimes0.2%29%20%5C%5C%20%20%5C%5C%20%3D0.196%2B0.183%2B0.16%2B0.12%3D0.659)
Part B:
The approximate the area of the region shown in the figure using the lower sums is given by:
![Area= [y(0)\times0.2]+[y(0.2)\times0.2]+[y(0.4)\times0.2]+[y(0.6)\times0.2] \\ +[y(0.8)\times0.2] \\ \\ =[\sqrt{1-(0)^2}\times0.2]+[\sqrt{1-(0.2)^2}\times0.2]+[\sqrt{1-(0.4)^2}\times0.2] \\ +[\sqrt{1-(0.6)^2}\times0.2] +[\sqrt{1-(0.8)^2}\times0.2] \\ \\ =(1\times0.2)+(0.9798\times0.2)+(0.9165\times0.2)+(0.8\times0.2)+(0.6\times0.2) \\ \\ =0.2+0.196+0.183+0.16+0.12=0.859](https://tex.z-dn.net/?f=Area%3D%20%5By%280%29%5Ctimes0.2%5D%2B%5By%280.2%29%5Ctimes0.2%5D%2B%5By%280.4%29%5Ctimes0.2%5D%2B%5By%280.6%29%5Ctimes0.2%5D%20%5C%5C%20%2B%5By%280.8%29%5Ctimes0.2%5D%20%5C%5C%20%5C%5C%20%3D%5B%5Csqrt%7B1-%280%29%5E2%7D%5Ctimes0.2%5D%2B%5B%5Csqrt%7B1-%280.2%29%5E2%7D%5Ctimes0.2%5D%2B%5B%5Csqrt%7B1-%280.4%29%5E2%7D%5Ctimes0.2%5D%20%5C%5C%20%2B%5B%5Csqrt%7B1-%280.6%29%5E2%7D%5Ctimes0.2%5D%20%2B%5B%5Csqrt%7B1-%280.8%29%5E2%7D%5Ctimes0.2%5D%20%5C%5C%20%5C%5C%20%3D%281%5Ctimes0.2%29%2B%280.9798%5Ctimes0.2%29%2B%280.9165%5Ctimes0.2%29%2B%280.8%5Ctimes0.2%29%2B%280.6%5Ctimes0.2%29%20%5C%5C%20%5C%5C%20%3D0.2%2B0.196%2B0.183%2B0.16%2B0.12%3D0.859)
Part C:
The approximate area of the given region is given by
Part A:
The approximate the area of the region shown in the figure using the lower sums is given by:
Part B:
The approximate the area of the region shown in the figure using the lower sums is given by:
Part C:
The approximate area of the given region is given by