4. (5 marks) In the previous assignment, you learned about the Gini index of a country (a number measuring the country's income inequality) is closely related to the area under the country's Lorenz curve over the interval [0, 1]. Let's recall that if the point (135, 11%) is on the Lorentz curve, then this means that the bottom (1% of households receive b% of the country's total income. In reality, we do not have a formula for the Lorenz curve but a collection of data. The following is data of Canadian households' (disposable) income in 2020. (https : //www150 . statcan . gc . ca/tl/tbll/en/tv. act ion?pid=36 10058701) Let's focus on the following 6 data points on the Lorenz curve LCanada- "Immun- _---|l So, for example, the point (0.2, 0.068) is on the Lorenz curve LCanada- This means that the bottom 20% of Canadian households received 6.8% of Canada's total (household disposable) income in 2020. (a) (1 mark) Use the data points above to approximate the area under the Lorenz curve 150,,de over the interval [0,1] by L5, the left endpoint approximation. (b) (1 mark) Use the data points above to approximate the area under the Lorenz curve 150,,de over the interval [0,1] by R5, the right endpoint approximation. (c) (2 marks) Use the data points above to apprOXimate the area under the Lorenz curve 13(36de over the interval [0,1] by T5, the Trapezoidal Rule. (d) (1 mark) It is known that Lorenz curves are always concave upward. Using this fact, determine if T5 is an underestimate or an overestimates. 5. Some integrals might seem complicated. But they will simplify signicantly if you look at them in the right way. Evaluate the integral _ cos3 (6) + sin(26) - 3cos(3) F(6) _ / sin3(6') + 3cos2 (6') + 2 sin(6') ._ 3 Hint: Try substituting u = sin(6')