LemonmathSelf-study sheet

Grade 12 · Lesson 4.2

Definite integrals

Name: Date: Time spent: min

Goals

Understand the meaning of a definite integral as the area under a graph and as a distance travelled.
Evaluate integrals with the formula $\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)$ .
Use the properties: constant factors, sums, swapping the limits and joining two adjacent intervals.
Evaluate integrals with an absolute value and integrals of functions of the form $f(ax+b)$ .

Remember

$\displaystyle\int_a^b f(x)\,dx=\Big[F(x)\Big]_a^b=F(b)-F(a)$ , where $F$ is any antiderivative of $f$ .
If $f\ge0$ , the integral is the area under the graph; if the velocity is positive, the integral is the distance travelled.
$\displaystyle\int_a^a=0$ , $\displaystyle\int_b^a=-\int_a^b$ , $\displaystyle\int_a^b=\int_a^c+\int_c^b$ ; constant factors and sums split apart.
With an absolute value: find where the sign changes, split the interval, remove the bars on each piece.
$\displaystyle\int f(ax+b)\,dx=\frac1aF(ax+b)+C$ .

Exercises

Exercise 1

Evaluate $\displaystyle\int_0^2(x^3-2x+3)\,dx$ .

Exercise 2

Evaluate $\displaystyle\int_1^4\left(\sqrt x-\frac{1}{x^2}\right)dx$ .

Exercise 3

Evaluate $\displaystyle\int_0^{\pi}(\sin x+2)\,dx$ .

Exercise 4

Given $\displaystyle\int_1^3 f(x)\,dx=5$ and $\displaystyle\int_1^3 g(x)\,dx=-2$ , find $\displaystyle\int_1^3\big(3f(x)-2g(x)+x\big)\,dx$ .

Exercise 5

Consider the integral $I=\displaystyle\int_0^3\lvert x-2\rvert\,dx$ . Is each claim true or false?

a) On $[0, 2]$ , $\lvert x-2\rvert=2-x$ .

b) $I=\left\lvert\displaystyle\int_0^3(x-2)\,dx\right\rvert=\dfrac32$ .

c) $I=\dfrac52$ .

d) $\displaystyle\int_3^0\lvert x-2\rvert\,dx=\dfrac52$ .

Exercise 6

Evaluate $\displaystyle\int_0^1(3x+1)^2\,dx$ .

Exercise 7

Water flows into a tank at the rate $r(t)=40-2t$ (litres per minute), $0\le t\le10$ . How much water flows into the tank in the first $10$ minutes?

Exercise 8 Review

$F(x)$ is an antiderivative of $f(x)$ on $\mathbb{R}$ with $F(1)=2$ and $F(4)=7$ . Find $\displaystyle\int_1^4 f(x)\,dx$ and $\displaystyle\int_1^4\big(2f(x)+1\big)\,dx$ .

Exercise 9 Review

Evaluate $\displaystyle\int_0^{\ln3}e^x\,dx$ and $\displaystyle\int_1^2 2^x\,dx$ (give the second one to 2 decimal places).

Exercise 10 Challenge

Evaluate $\displaystyle\int_{-1}^2\lvert x^2-x\rvert\,dx$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The integral

\displaystyle\int_1^2(2x+1)\,dx

equals

6

.B.

-4

.C.

4

.D.

2

Question 2. Given

\displaystyle\int_0^5 f(x)\,dx=7

and

\displaystyle\int_0^2 f(x)\,dx=3

, the value of

\displaystyle\int_2^5 f(x)\,dx

10

.B.

4

.C.

-4

.D.

-10

Question 3. The integral

\displaystyle\int_0^{\pi}\sin x\,dx

equals

-2

.B.

1

.C.

2

.D.

0

Question 4. The integral

\displaystyle\int_0^1 e^{2x}\,dx

equals

e^2-1

.B.

\dfrac{e^2-1}{2}

.C.

2(e^2-1)

.D.

\dfrac{e^2}{2}

Part II. True or false

Question 1. Consider

f(x)=x^2-4x+3

a) $\displaystyle\int_0^3 f(x)\,dx=0$ .	T / F
b) $f(x)\le0$ for every $x\in[1, 3]$ .	T / F
c) $\displaystyle\int_0^3\lvert f(x)\rvert\,dx=0$ .	T / F
d) $\displaystyle\int_0^3\lvert f(x)\rvert\,dx=\dfrac83$ .	T / F

Part III. Short answer

Question 1. Evaluate

\displaystyle\int_0^2(x^3+3x^2)\,dx

Question 2. Evaluate

\displaystyle\int_1^e\left(2x+\frac1x\right)dx

. Give your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)$ .

Steps:

Find an antiderivative $F(x)=\dfrac{x^4}{4}-x^2+3x$ .
Compute $F(2)-F(0)$ .

Exercise 2.

What you need: $\sqrt x=x^{\frac12}$ has antiderivative $\dfrac23x\sqrt x$ ; $x^{-2}$ has antiderivative $-\dfrac1x$ .

Steps:

Write an antiderivative $F(x)=\dfrac23x\sqrt x+\dfrac1x$ .
Compute $F(4)-F(1)$ , taking care with the fractions.

Exercise 3.

What you need: the antiderivative of $\sin x$ is $-\cos x$ ; $\cos\pi=-1$ and $\cos0=1$ .

Steps:

Write $F(x)=-\cos x+2x$ .
Compute $F(\pi)-F(0)$ .

Exercise 4.

What you need: split the sum and take constants outside the integral.

Steps:

Write it as $3\cdot5-2\cdot(-2)+\displaystyle\int_1^3x\,dx$ .
Evaluate $\displaystyle\int_1^3x\,dx$ .

Exercise 5.

What you need: split the interval where $x-2$ changes sign; swapping the limits changes the sign.

Steps:

Find the sign of $x-2$ on $[0, 2]$ and on $[2, 3]$ .
Evaluate the two smaller integrals and add.

Exercise 6.

What you need: $\displaystyle\int f(ax+b)\,dx=\frac1aF(ax+b)+C$ , here with $a=3$ .

Steps:

An antiderivative is $\dfrac{(3x+1)^3}{9}$ .
Substitute the limits $1$ and $0$ .

Exercise 7.

What you need: the amount that flows in from $t=a$ to $t=b$ is $\displaystyle\int_a^b r(t)\,dt$ (the rate is non-negative).

Steps:

Write $\displaystyle\int_0^{10}(40-2t)\,dt$ .
Evaluate it with the antiderivative $40t-t^2$ .

Exercise 8.

What you need: the definition of the integral through an antiderivative (see antiderivatives).

Steps:

$\displaystyle\int_1^4f(x)\,dx=F(4)-F(1)$ .
Split the sum; $\displaystyle\int_1^41\,dx=3$ .

Exercise 9.

What you need: $e^{\ln a}=a$ (logarithms, grade 11); the antiderivative of $2^x$ is $\dfrac{2^x}{\ln2}$ .

Steps:

$\Big[e^x\Big]_0^{\ln3}=e^{\ln3}-e^0$ .
$\Big[\dfrac{2^x}{\ln2}\Big]_1^2=\dfrac{4-2}{\ln2}$ .

Exercise 10.

What you need: $x^2-x=x(x-1)$ changes sign at $0$ and $1$ ; it is positive outside the roots and negative between them.

Steps:

Split into the three intervals $[-1, 0]$ , $[0, 1]$ , $[1, 2]$ .
Use the antiderivative $G(x)=\dfrac{x^3}{3}-\dfrac{x^2}{2}$ on each piece, changing the sign on the middle one.

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