ATHS Calculus Project Grade 12 ES Done by

ATHS Calculus Project Grade: 12 – ES Done by : Grade 12 -01 , ID: Ali Al-Ali 1001299 Majid Al Yammahi 1001330 Mohammed Saeed 1001293 Ahmem Al Naqbi 1001391 S

GOALS S Students will interpret their understanding of applications of integration S Students will use technology for research and communication: wiki/blog. S Students will present their findings to the class through digital Media(Video/Booklet).

Theme /INTRO S This project covers the application of topics in calculus; students will apply the concepts and theorems they have learned on integration. Moreover students will be able to apply their knowledge in relation to vari

TASK 1 S The boundaries of a piece of land that belongs to your family can be best described as in the following graph: S i. e. the area between the ��−axis and the graph of the function �� (��)=�� 3 − 55�� 2 +6��+

S Answer the following questions: S 1. Using approximating rectangles dividing the interval from 0 to 4 into 5 equal subdivisions, estimate the area between f(��)= �� 3 − 5� and the ��−axis, with x=0. 8. ∆

S 2. Calculate the exact value of the same area using the definite integral.

Task 2 ”Oil Consumption” S Suppose that the rate of consumption of a natural resource is ��′(��), where: ��′(��) = ����. S Here �� is time in years, �� is a constant, and �� is the consumption in the year when �� = 0. In 2006, an oil company sold 1. 2 billion barrels of oil. Assume that ��=0. 04.

S b) Set up a definite integral for the amount of oil that the company will sell in the next 10 years.

S c) Evaluate the definite integral of part b.

S d) The company has about 20 billions barrels of oil in reserve. To find the number of years that this amount will last, solve the equation:

S e) Rework part d, assuming that �� = 0. 02

Task 3 S The annual revenue for your company over the past five years was found to follow the following Integral formula: Where n is the number of years

S 1. Verify the following relation between two consecutive years' revenue:

S 2. Find the revenue of year 2, i. e. Find U 2

Task 4 S A water well of depth 50 feet is used to supply people with water. A rope of weight 0. 2 lbs/ft is used take the water out of the well using a barrel that weight 80 lbs when it is full of water.

S a) Calculate the work needed to lift the barrel from the bottom to the midpoint of the well.

S b) Calculate the work needed to lift the barrel from the midpoint of the well to the top of the well.

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