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Subsection 1.2.3 Exercises

Exercises — Stage 1

1

For each of the following properties of definite integrals, draw a picture illustrating the concept, interpreting definite integrals as areas under a curve.

For simplicity, you may assume that \(a \leq c \leq b\text{,}\) and that \(f(x),g(x)\) give positive values.

  1. \(\displaystyle\int_a^a f(x)\,\dee{x}=0\text{,}\) (Theorem 1.2.3, part (a))
  2. \(\displaystyle\int_a^b f(x)\,\dee{x}= \displaystyle\int_a^c f(x)\,\dee{x} + \int_c^b f(x)\dee{x} \text{,}\) (Theorem 1.2.3, part (c))
  3. \(\displaystyle\int_a^b \left( f(x) + g(x) \right)\,\dee{x} = \displaystyle\int_a^b f(x)\,\dee{x} + \displaystyle\int_a^b g(x)\,\dee{x}\text{,}\) (Theorem 1.2.1, part (a))
2

If \(\displaystyle\int_0^b \cos x\dee{x}=\sin b\text{,}\) then what is \(\displaystyle\int_a^b \cos x\dee{x}\text{?}\)

3 (✳)

Decide whether each of the following statements is true or false. If false, provide a counterexample. If true, provide a brief justification. (Assume that \(f(x)\) and \(g(x)\) are continuous functions.)

  1. \(\displaystyle\int_{-3}^{-2} f(x) \dee{x}=-\displaystyle\int_{3}^{2} f(x) \dee{x}\text{.}\)
  2. If \(f(x)\) is an odd function, then \(\displaystyle \int_{-3}^{-2} f(x)\,\dee{x} = \int_2^3 f(x)\,\dee{x}\text{.}\)
  3. \(\displaystyle\int_{0}^{1} f(x)\cdot g(x) \dee{x} =\int_{0}^{1} f(x) \dee{x} \cdot \int_{0}^{1} g(x)\dee{x}\text{.}\)
4

Suppose we want to make a right Riemann sum with 100 intervals to approximate \(\int\limits_5^0 f(x)\dee{x}\text{,}\) where \(f(x)\) is a function that gives only positive values.

  1. What is \(\Delta x\text{?}\)
  2. Are the heights of our rectangles positive or negative?
  3. Is our Riemann sum positive or negative?
  4. Is the signed area under the curve \(y=f(x)\) from \(x=0\) to \(x=5\) positive or negative?

Exercises — Stage 2

5 (✳)

Suppose \(\displaystyle\int_2^3 f(x)\,\dee{x} = -1\) and \(\displaystyle\int_2^3 g(x)\,\dee{x} = 5\text{.}\) Evaluate \(\displaystyle \int_2^3 \big( 6 f(x) - 3 g(x) \big)\,\dee{x}\text{.}\)

6 (✳)

If \(\displaystyle\int_0^2 f(x)\,\dee{x} = 3\) and \(\displaystyle\int_0^2 g(x)\,\dee{x} = -4\text{,}\) calculate \(\displaystyle \int_0^2 \big( 2 f(x) + 3 g(x) \big)\,\dee{x}\text{.}\)

7 (✳)

The functions \(f(x)\) and \(g(x)\) obey

\begin{align*} \int_0^{-1} f(x)\,\dee{x} &= 1 & \int_0^2 f(x)\,\dee{x} &= 2 \\ \int_{-1}^0 g(x)\,\dee{x} &= 3 & \int_0^2 g(x)\,\dee{x} &= 4 \end{align*}

Find \(\int_{-1}^2 \big[3g(x)-f(x)\big]\,\dee{x}\text{.}\)

8

In Question 1.1.8.45, Section 1.1, we found that

\begin{equation*} \int_0^a\sqrt{1-x^2}\dee{x}=\frac{\pi}{4} - \frac{1}{2}\arccos(a)+\frac{1}{2}a\sqrt{1-a^2} \end{equation*}

when \(0\le a\le 1\text{.}\)

Using this fact, evaluate the following:

  1. \(\displaystyle\int_{a}^0 \sqrt{1-x^2}\dee{x}\text{,}\) where \(-1 \leq a \leq 0\)
  2. \(\displaystyle\int_{a}^1 \sqrt{1-x^2}\dee{x}\text{,}\) where \(0 \leq a \leq 1\)
9 (✳)

Evaluate \({\displaystyle\int_{-1}^2 |2x|\dee{x}}\text{.}\)

You may use the result from Example 1.2.6 that \(\int\limits_a^b x\dee{x}=\frac{b^2-a^2}{2} \text{.}\)

10

Evaluate \(\displaystyle\int_{-5}^5 x|x|\dee{x}\,.\)

11

Suppose \(f(x)\) is an even function and \(\displaystyle\int_{-2}^2 f(x)\dee{x}=10\text{.}\) What is \(\displaystyle\int_{-2}^0 f(x)\dee{x}\text{?}\)

Exercises — Stage 3

12 (✳)

Evaluate \(\displaystyle\int_{-2}^{2} \left(5+\sqrt{4-x^2}\right)\dee{x}\text{.}\)

13 (✳)

Evaluate \(\displaystyle\int_{-2012}^{+2012} \frac{\sin x}{\log(3+x^2)}\dee{x}\text{.}\)

14 (✳)

Evaluate \(\displaystyle\int_{-2012}^{+2012} x^{1/3}\cos x\,\dee{x}\text{.}\)

15

Evaluate \(\displaystyle\int_{0}^6 (x-3)^3\,\dee{x}\,.\)

16

We want to compute the area of an ellipse, \((ax)^2+(by)^2=1\) for some (let's say positive) constants \(a\) and \(b\text{.}\)

  1. Solve the equation for the upper half of the ellipse. It should have the form “\(y=\cdots\)”
  2. Write an integral for the area of the upper half of the ellipse. Using properties of integrals, make the integrand look like the upper half of a circle.
  3. Using geometry and your answer to part (b), find the area of the ellipse.
17

Fill in the following table: the product of an (even/odd) function with an (even/odd) function is an (even/odd) function. You may assume that both functions are defined for all real numbers.

\(\times\) even odd
even
odd
18

Suppose \(f(x)\) is an odd function and \(g(x)\) is an even function, both defined at \(x=0\text{.}\) What are the possible values of \(f(0)\) and \(g(0)\text{?}\)

19

Suppose \(f(x)\) is a function defined on all real numbers that is both even and odd. What could \(f(x)\) be?

20

Is the derivative of an even function even or odd? Is the derivative of an odd function even or odd?

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