Review: integral calculus

Integral calculus, grade 11 (5 units) Lesson 20 of 20Practice

Review: integral calculus

What the course taught

The antiderivative: $\displaystyle {\int f(x)\,dx}={F(x)+C}$ , where $\displaystyle {F'}={f}$ ; powers ( $\displaystyle n\neq-1$ ), roots, $\displaystyle \sin$ , $\displaystyle \cos$ , $\displaystyle \frac{1}{\cos^2x}$ .
Rules: a constant factor and a sum; $\displaystyle {\int f(mx+b)\,dx}={\frac{1}{m}F(mx+b)+C}$ ; $\displaystyle {\int f(u(x))u'(x)\,dx}={F(u(x))+C}$ ; identities before integrating.
Accumulation: the accumulated quantity is the area under the graph of the rate, with its sign; $\displaystyle {F_a(x)}={\int_a^xf(t)\,dt}$ .
The fundamental theorem: $\displaystyle {F_a'(x)}={f(x)}$ , so $\displaystyle {\int_a^bf(x)\,dx}={F(b)-F(a)}$ .
Graphs: $\displaystyle f$ positive – $\displaystyle F$ rises; $\displaystyle f$ changes sign – an extremum of $\displaystyle F$ ; an extremum of $\displaystyle f$ – an inflection point of $\displaystyle F$ .
Applications: a function from its derivative and a point; an area – split at the zeros and the meeting points, the upper minus the lower; displacement and distance; a parameter and an extremum.

The function $\displaystyle {f(x)}={x-\frac{\sin2x}{2}}$ is given.

Show that $\displaystyle {f'(x)}={2\sin^2x}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Does $\displaystyle f$ have extremum points? Explain.

The function $\displaystyle {g(x)}={x+\sin^2x}$ is given, for $\displaystyle -\pi\le x\le\pi$ .
Calculate the area bounded by the graph of $\displaystyle g$ and the line $\displaystyle {y}={x}$ .

\displaystyle g(x)=x+\sin^{2}x

\displaystyle y=x

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The board below shows the graph of $\displaystyle {f(x)}={x(x-2)^2}$ .
$\displaystyle {F(x)}={\int_0^xf(t)\,dt}$ .

Where does $\displaystyle F$ rise, and where does it fall? Explain from the graph of $\displaystyle f$ .

\displaystyle f(x)=x(x-2)^{2}

Write $\displaystyle F(x)$ without an integral.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Mark the extremum points of $\displaystyle F$ on the board.

\displaystyle f(x)=x(x-2)^{2}

Mark the inflection points of $\displaystyle F$ on the board.

\displaystyle f(x)=x(x-2)^{2}

$\displaystyle {f'(x)}={(2x-1)^2}$ .
The line $\displaystyle {y}={x+1}$ is tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={1}$ .

Find all the antiderivatives of $\displaystyle f'$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find $\displaystyle f(x)$ , and write it on the board.

\displaystyle y=x+1

The function $\displaystyle {f(x)}={\sqrt{x}}$ is given.
The line $\displaystyle {y}={\frac{x}{4}+1}$ is tangent to the graph of $\displaystyle f$ at the point $\displaystyle (4,2)$ .

Calculate the area bounded by the graph of $\displaystyle f$ , the tangent and the $\displaystyle y$ -axis.

\displaystyle f(x)=\sqrt{x}

\displaystyle g(x)=\frac{x}{4}+1

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: integral calculus, grade 11 (5 units)

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Basic integrals

The indefinite integral – all the antiderivatives
$\displaystyle \int f(x)\,dx=F(x)+C$ $\displaystyle F^{\prime}(x)=f(x)$
A power
$\displaystyle \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
A constant
$\displaystyle \int k\,dx=kx+C$
$\displaystyle \int\frac{1}{x^{2}}\,dx=-\frac{1}{x}+C$
One over a root
$\displaystyle \int\frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C$
A root
$\displaystyle \int\sqrt{x}\,dx=\frac{2}{3}x\sqrt{x}+C$

The rules of integration

A constant factor
$\displaystyle \int k\cdot f(x)\,dx$ $\displaystyle =$ $\displaystyle k\int f(x)\,dx$
A sum and a difference
$\displaystyle \int\left(f(x)\pm g(x)\right)dx$ $\displaystyle =$ $\displaystyle \int f(x)\,dx\pm\int g(x)\,dx$
A linear inner function
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
A composite function times the inner derivative
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
A power of a function times its derivative
$\displaystyle \int\left(u(x)\right)^{n}\cdot u^{\prime}(x)\,dx=\frac{\left(u(x)\right)^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$

Trigonometric functions (in radians)

$\displaystyle \int\cos x\,dx=\sin x+C$
$\displaystyle \int\sin x\,dx=-\cos x+C$
$\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C$
From $\displaystyle {\cos2x}={1-2\sin^2x}$
$\displaystyle \sin^{2}x$ $\displaystyle =$ $\displaystyle \frac{1-\cos2x}{2}$
From $\displaystyle {\cos2x}={2\cos^2x-1}$
$\displaystyle \cos^{2}x$ $\displaystyle =$ $\displaystyle \frac{1+\cos2x}{2}$
From $\displaystyle {\sin2x}={2\sin x\cos x}$
$\displaystyle \sin x\cos x$ $\displaystyle =$ $\displaystyle \frac{1}{2}\sin2x$
From $\displaystyle {\sin^2x+\cos^2x}={1}$
$\displaystyle \tan^{2}x$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}x}-1$

The accumulation function and the definite integral

The accumulation function of $\displaystyle f$ from $\displaystyle a$
$\displaystyle F_{a}(x)=\int_{a}^{x}f(t)\,dt$
The fundamental theorem
$\displaystyle F_{a}^{\prime}(x)=f(x)$
The definite integral, $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The same limit
$\displaystyle \int_{a}^{a}f(x)\,dx=0$
Swapping the limits
$\displaystyle \int_{b}^{a}f(x)\,dx$ $\displaystyle =$ $\displaystyle -\int_{a}^{b}f(x)\,dx$
Joining intervals
$\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx$ $\displaystyle =$ $\displaystyle \int_{a}^{c}f(x)\,dx$

Areas and accumulation

An area above the $\displaystyle x$ -axis
$\displaystyle S=\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\geq0$
An area below the $\displaystyle x$ -axis
$\displaystyle S=-\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\leq0$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The displacement from the velocity
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$