LemonmathSelf-study sheet

Grade 12 · Lesson 7.4

Revision: antiderivatives and integrals

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Goals

Find antiderivatives using the table of standard results and the properties, including an antiderivative satisfying a given condition.
Evaluate definite integrals with the Newton–Leibniz formula and use the properties of integrals.
Find the area of a plane region and the volume of a solid of revolution about the $x$ -axis.
Use integrals in real-life problems: distance from velocity, total change from a rate of change.

Exercises

Exercise 1 Multiple choice

$\displaystyle\int\left(4x^3-\frac{1}{x^2}\right)dx$ equals A. $x^4+\dfrac1x+C$ . B. $x^4-\dfrac1x+C$ . C. $12x^2+\dfrac{2}{x^3}+C$ . D. $x^4+\dfrac{1}{3x^3}+C$ .

Exercise 2 Multiple choice

$\displaystyle\int_0^{\frac{\pi}{2}}\cos x\,dx$ equals A. $1$ . B. $0$ . C. $-1$ . D. $\dfrac{\pi}{2}$ .

Exercise 3 Short answer

Evaluate $\displaystyle\int_0^2\left(3x^2-2x+1\right)dx$ .

Exercise 4 True/False

Let $f(x)=e^x+2x$ .

a) $F(x)=e^x+x^2$ is an antiderivative of $f(x)$ .

b) $\displaystyle\int_0^1 f(x)\,dx=e+1$ .

c) The antiderivative $F$ of $f$ with $F(0)=3$ is $F(x)=e^x+x^2+3$ .

d) $\displaystyle\int_{-1}^1 f(x)\,dx=e-\dfrac1e$ .

Exercise 5 Short answer

Find the area of the region bounded by the parabola $y=x^2-4$ and the $x$ -axis. Give your answer to 1 decimal place.

Exercise 6 Written answer

Find the area of the region bounded by the graph of $y=x^3-x$ , the $x$ -axis and the lines $x=0$ , $x=2$ .

Exercise 7 Multiple choice

Rotating the region bounded by $y=2x$ , the $x$ -axis and the line $x=3$ about the $x$ -axis gives a cone of volume A. $36\pi$ . B. $9\pi$ . C. $108\pi$ . D. $18\pi$ .

Exercise 8 Short answer

Water flows into a tank at a rate of $r(t)=0.6t+2$ (litres per minute), with $t$ in minutes. How many litres flow into the tank in the first $10$ minutes?

Exercise 9 True/False

A car brakes and slows down uniformly with velocity $v(t)=12-3t$ (m/s), where $t$ is the time in seconds since braking.

a) The car stops after $4$ seconds.

b) The distance travelled $t$ seconds after braking is $s(t)=12t-1.5t^2$ (m), $0\le t\le4$ .

c) From braking to stopping, the car travels $24$ m.

d) In the last second before stopping, the car travels $3$ m.

Exercise 10 Challenge

A gateway has the shape of a parabola: it is $8$ m wide at ground level and its top is $4$ m above the ground. The whole gateway is filled with glass. Find the area of glass, in m², to 1 decimal place.

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 family of antiderivatives of

f(x)=2^x

\dfrac{2^x}{\ln2}+C

.B.

2^x\ln2+C

.C.

\dfrac{2^{x+1}}{x+1}+C

.D.

x\cdot2^{x-1}+C

Question 2.

\displaystyle\int_1^2\frac1x\,dx

equals

\ln2

.B.

\dfrac34

.C.

\ln3

.D.

\dfrac12

Question 3. The functions

f

g

are continuous on

[a, b]

. The area of the region bounded by

y=f(x)

y=g(x)

x=a

x=b

\displaystyle\int_a^b\lvert f(x)-g(x)\rvert\,dx

.B.

\displaystyle\int_a^b\left[f(x)-g(x)\right]dx

.C.

\pi\displaystyle\int_a^b\left[f(x)-g(x)\right]^2dx

.D.

\left\lvert\displaystyle\int_a^b\left[f(x)-g(x)\right]dx\right\rvert

Question 4. Given

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

, the value of

\displaystyle\int_0^2\left[2f(x)+x\right]dx

8

.B.

7

.C.

6

.D.

10

Part II. True or false

Question 1. The region

H

is bounded by

y=x^2

, the

x

-axis and the lines

x=0

x=3

a) The area of $H$ is $S=\displaystyle\int_0^3x^2\,dx$ .	T / F
b) $S=9$ .	T / F
c) The volume of the solid formed by rotating $H$ about the $x$ -axis is $V=\pi\displaystyle\int_0^3x^2\,dx$ .	T / F
d) $V=\dfrac{243\pi}{5}$ .	T / F

Part III. Short answer

Question 1. An object moves with velocity

v(t)=3t^2+2t

(m/s). Find the distance it travels from

t=0

t=4

seconds, in metres.

Question 2. Evaluate

\displaystyle\int_0^1\left(e^x+1\right)dx

. Give your answer to 2 decimal places.

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

Hints (only when stuck)

Exercise 1.

What you need: $\dfrac{1}{x^2}=x^{-2}$ and $\displaystyle\int x^{\alpha}\,dx=\dfrac{x^{\alpha+1}}{\alpha+1}+C$ .

Steps:

$\displaystyle\int x^{-2}\,dx=\dfrac{x^{-1}}{-1}=-\dfrac1x$ .
Watch the minus sign in front.

Exercise 2.

What you need: an antiderivative of $\cos x$ is $\sin x$ .

Steps:

Work out $\sin\dfrac{\pi}{2}-\sin0$ .

Exercise 3.

What you need: the Newton–Leibniz formula.

Steps:

An antiderivative is $x^3-x^2+x$ .
Substitute the limits $2$ and $0$ .

Exercise 4.

What you need: check an antiderivative by differentiating; $e^0=1$ .

Steps:

Write $F(x)=e^x+x^2+C$ .
Evaluate each integral with $\Big[e^x+x^2\Big]$ .

Exercise 5.

What you need: $S=\displaystyle\int_a^b\lvert f(x)\rvert\,dx$ , where $a$ , $b$ are the intersections with the $x$ -axis.

Steps:

Solve $x^2-4=0$ .
On $[-2, 2]$ , $x^2-4\le0$ , so $\lvert x^2-4\rvert=4-x^2$ .

Exercise 6.

What you need: split the interval at the roots of $f(x)$ to remove the modulus.

Steps:

$x^3-x=x(x-1)(x+1)$ : negative on $(0, 1)$ , positive on $(1, 2)$ .
Evaluate the two integrals and add.

Exercise 7.

What you need: $V=\pi\displaystyle\int_a^b f^2(x)\,dx$ , here with $f^2(x)=4x^2$ .

Steps:

Work out $\pi\displaystyle\int_0^3 4x^2\,dx$ .
Check with the cone formula $\dfrac13\pi r^2h$ .

Exercise 8.

What you need: the amount of water is $\displaystyle\int_0^{10} r(t)\,dt$ .

Steps:

An antiderivative is $0.3t^2+2t$ .
Substitute the limits.

Exercise 9.

What you need: $s(t)=\displaystyle\int_0^t v(u)\,du$ ; the distance from $t=a$ to $t=b$ is $s(b)-s(a)$ .

Steps:

Solve $v(t)=0$ .
Work out $s(4)$ and $s(4)-s(3)$ .

Exercise 10.

What you need: put the origin at the middle of the base with the $y$ -axis through the top; the area is $\displaystyle\int_{-4}^4 f(x)\,dx$ .

Steps:

The parabola passes through $(-4, 0)$ , $(4, 0)$ , $(0, 4)$ : $y=4-\dfrac{x^2}{4}$ .
Evaluate the integral.

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