Using the Definition

\[\int_a^b f(x) dx = R(n) = \lim n \rightarrow \infty (\frac{b-a}{n})\sum_{i=1}^n f(a + i\frac{b-a}{n})\]

import sympy as sy

x, n = sy.symbols('x n')

sy.pprint(sy.summation(x, (x, 1,n)))

 2
n    n
── + ─
2    2

sy.pprint(sy.summation(x**2, (x, 1,n)))

 3    2
n    n    n
── + ── + ─
3    2    6

sy.pprint(sy.summation(x**3, (x, 1,n)))

 4    3    2
n    n    n
── + ── + ──
4    2    4

sy.pprint(sy.summation(x**4, (x, 1,n)))

 5    4    3
n    n    n    n
── + ── + ── - ──
5    2    3    30

Problem I

Use the definition and our summation formulas to evaluate the area under the given function and approximate what happens as \(n \rightarrow \infty\) on given domain:

1. \(f(x) = x\) on \([1,4]\)
2. \(g(x) = x^2 - 2x\) on \([2, 4]\)
3. \(h(x) = x^3 - x + 1\) on \([1, 3]\)

Problem II

We want to examine patterns on the interval \([0,b]\) for polynomial functions. Let’s use the definition to prove the following:

1. \(\int_0^b x = \frac{b^2}{2}\)
2. \(\int_0^b x^2 = \frac{b^3}{3}\)
3. \(\int_0^b x^3 = \frac{b^4}{4}\)

Problem III

Theorem: Assume that \(f(x)\) is continuous on \([a,b]\) and let \(F(x)\) be an antiderivative of \(f(x)\) on \([a,b]\). Then

\[\int_a^b f(x) = F(b) - F(a)\]

Use the attached table of integrals and the theorem above to evaluate the following definite integrals.

1. \(\int_0^2 2x^2 - x ~ dx\)
2. \(\int_{1/2}^{2} \ln{x} ~ dx\)
3. \(\int_0^{2\pi} \sin{x} ~ dx\)

Problem IV

Interpreting the Integral as Total Change:

Water flows into an empty bucket at a rate of \(r(t)\) gallons per second. How much water is in the bucket after 5 seconds? If the rate is constant, we would have \(0.3 \times 5 = 1.5\) gallons. If the rate is not constant, we can interpret the quantity of water as equal to the area under the graph of \(r(t)\).

1. A survey shows that a mayoral candidate is gaining votes at a rate of \(2000t + 150\) votes per day, where \(t\) is the number of days since announcing her candidacy. How many supporters after 90 days?
2. A projectile is released with initial (vertical) velocity 100 m/s. Use the formula \(v(t) = 100 - 98t\) for velocity to determine the distance traveled during the first 15 seconds.
3. The rate at which water drains from a tank is recorded at half-minute intervals. Use approximations to estimate the total amount of water drained during the first 3 minutes.

liters	\(t\)(min)
0	50
0.5	48
1	46
1.5	43
2	40
2.5	39
3	36

Problem V

Area Between Curves

Use the definite integral to find specified area between two curves.

1. Area between \(y = x^3 - 2x^2 + 10\) and \(y = 3x^2 + 4x - 10\).
2. Area between \(y = 0.5 x\) and \(y = x\sqrt{1 - x^2}\)
3. Area between \(y = 4 - x^2\) and \(y = x^2 - 4\)