Try It Yourself#
Exercise 1#
Area between two curves on given interval
Compute the area of the region bounded by \(y=x+1\) and \(y=x^2-3x+1\) on the interval \([1,3]\).
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Answer: \(22/3\)
Exercise 2#
Area between two curves on given interval
Compute the area of the region bounded by \(y=x+1\) and \(y=x^2-3x+1\) on the interval \([-1,1]\).
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Answer: \(4\)
Exercise 3#
Area between two curves
Compute the area of the region bounded by \(y=x+1\) and \(y=x^2-3x+1\).
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Answer: \(32/3\)
Exercise 4#
Area between two curves
Compute the area of the region bounded by \(y = x^8\) and \(y = x^4\).
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Answer: \(8/45\)
Exercise 5#
Area between two curves on given interval
Compute the area of the region bounded by \(y = x^4 + x^3 + 1\) and \(y = x^4+1\) on the interval \([3,4]\).
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Answer: \(175/4\)
Exercise 6#
Area between two curves on given interval
Compute the area of the region bounded by \(y = x^2 - x\) and \(y = x\) on the interval \([-2, 2]\).
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Answer: \(8\)
Exercise 7#
Consumer surplus at market equilibrium
If the demand function for math anxiety pills is \(p = D(x) = 21-5x\) and the corresponding supply function is \(p=S(x) = 3x+5\), determine the consumer surplus at the market equilibrium point.
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Answer: \(10\)
Exercise 8#
Consumer surplus
If the demand function for math anxiety pills is \(p=D(x)=\sqrt{16-2x}\), determine the consumer surplus at the market price of $2.
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Answer: \(20/3\)
Exercise 9#
Producer surplus
If the supply function for a brand of a math magazines is given by \(p = S(x) = 50 + 80e^{0.05x}\), determine the producer surplus at the market price of $530.
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Answer: \(9600\ln(6)-8000\)
Exercise 10#
Future value of an income stream
A math t-shirt business is expected to generate $40,000 in revenue per year for the next 25 years. If the income is reinvested in the business at a rate of 2% per year compounded continuously, what is the future value of this income stream at the end of 25 years?
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Answer: \(\displaystyle e^{0.5}\int_0^{25}40,000e^{-0.02t} ~dt = 2,000,000e^{0.5}(1-e^{-0.5})\)
Exercise 11#
Future value of an income stream
A Math 110 student decides to make semiannual payments of $2,000 into a retirement account paying 3% interest per year compounded continuously. How much will the student have in their retirement account after 40 years?
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Answer: \(\dfrac{4,000}{0.03}(e^{1.2}-1)\)