Practice: accumulation and the definite integral

Integral calculus, grade 11 (5 units) Lesson 8 of 20Practice

Practice: accumulation and the definite integral

Find the integrals.

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A pump fills a pool.
In the first two hours it pumps 30 m³ an hour, in the next three hours 50 m³ an hour, and then it stops for an hour.
In the seventh hour 20 m³ are let out of the pool.
At the start the pool was empty.

Sketch the graph of the rate of change of the amount of water.

How much water is in the pool after 7 hours?

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The function $\displaystyle {f(x)}={\frac{1}{x}}$ is given for $\displaystyle 1\le x\le3$ .
The interval is split into four pieces, each $\displaystyle 0.5$ wide.

Calculate the sum of the rectangles by the left end of each piece.

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$\displaystyle f$ falls. Is the sum more or less than the exact area? Explain.

The board below shows the graph of $\displaystyle f$ .
$\displaystyle {F(x)}={\int_{-1}^x f(t)\,dt}$ .

\displaystyle f(x)

Calculate $\displaystyle F(0)$ , $\displaystyle F(2)$ , $\displaystyle F(3)$ and $\displaystyle F(4)$ .

For which $\displaystyle x$ in $\displaystyle -1\le x\le4$ is the value of $\displaystyle F$ greatest? What is it?

\displaystyle x

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The greatest value

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Over which interval is $\displaystyle F$ a linear function? Explain.

$\displaystyle {F(x)}={\int_2^x\left(t^2-4\right)dt}$ .

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

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

\displaystyle f(x)=x^{2}-4

Calculate the integrals.

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Given: $\displaystyle {\int_1^4 g(x)\,dx}={5}$ .
Calculate:

$\displaystyle \int_1^4\left(3g(x)-2\right)dx$

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$\displaystyle \int_4^1\left(g(x)+x\right)dx$

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$\displaystyle a>1$ is a number for which $\displaystyle {\int_1^a\frac{2}{x^2}\,dx}={1}$ .

Calculate the integral as an expression in $\displaystyle a$ .

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Find $\displaystyle a$ .

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
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$