2.1 Övningar
Aus Förberedande kurs i matematik 2
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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | ||
| style="border-bottom:1px solid #000" width="5px" | | | style="border-bottom:1px solid #000" width="5px" | | ||
| - | {{ | + | {{Ej vald flik|[[2.1 Inledning till integraler|Theory]]}} |
| - | {{ | + | {{Vald flik|[[2.1 Exercises|Exercises]]}} |
| style="border-bottom:1px solid #000" width="100%"| | | style="border-bottom:1px solid #000" width="100%"| | ||
|} | |} | ||
| - | === | + | ===Exercise 2.1:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Interpret each integral as an area, and determine its value. | |
| + | |||
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 23: | Zeile 24: | ||
</div>{{#NAVCONTENT:Svar|Svar 2.1:1|Lösning a|Lösning 2.1:1a|Lösning b|Lösning 2.1:1b|Lösning c|Lösning 2.1:1c|Lösning d|Lösning 2.1:1d}} | </div>{{#NAVCONTENT:Svar|Svar 2.1:1|Lösning a|Lösning 2.1:1a|Lösning b|Lösning 2.1:1b|Lösning c|Lösning 2.1:1c|Lösning d|Lösning 2.1:1d}} | ||
| - | === | + | ===Exercise 2.1:2=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Calculate the integrals | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 37: | Zeile 38: | ||
|width="50%"| <math>\displaystyle\int_{1}^{4} \displaystyle\frac{\sqrt{x}}{x^2}\, dx</math> | |width="50%"| <math>\displaystyle\int_{1}^{4} \displaystyle\frac{\sqrt{x}}{x^2}\, dx</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Svar|Svar 2.2:1|Lösning | + | </div>{{#NAVCONTENT:Svar|Svar 2.1:2|Lösning a|Lösning 2.1:2a|Lösning b|Lösning 2.1:2b|Lösning c|Lösning 2.1:2c|Lösning d|Lösning 2.1:2d}} |
| + | |||
| + | ===Exercise 2.1:3=== | ||
| + | <div class="ovning"> | ||
| + | Calculate the integrals | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="50%"|<math>\displaystyle\int \sin x\, dx</math> | ||
| + | |b) | ||
| + | |width="50%"| <math>\displaystyle\int 2\sin x \cos x\, dx</math> | ||
| + | |- | ||
| + | |c) | ||
| + | |width="50%"| <math> \displaystyle\int e^{2x}(e^x+1)\, dx</math> | ||
| + | |d) | ||
| + | |width="50%"| <math>\displaystyle\int \displaystyle\frac{x^2+1}{x}\, dx</math> | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 2.1:3|Lösning a|Lösning 2.1:3a|Lösning b|Lösning 2.1:3b|Lösning c|Lösning 2.1:3c|Lösning d|Lösning 2.1:3d}} | ||
| + | |||
| + | ===Exercise 2.1:4=== | ||
| + | <div class="ovning"> | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="100%"| Calculate the area between the curve <math>y=\sin x</math> and the <math>x</math>-axis when <math>0\le x \le \frac{5\pi}{4}</math>. | ||
| + | |- | ||
| + | |b) | ||
| + | |width="100%"| Calculate the area under the curve <math>y=-x^2+2x+2</math> and above the <math>x</math>-axis. | ||
| + | |- | ||
| + | |c) | ||
| + | |width="100%"| Calculate the area of the finite region between the curves <math> y=\frac{1}{4}x^2+2</math> and <math>y=8-\frac{1}{8}x^2</math> (Swedish A-level 1965). | ||
| + | |- | ||
| + | |d) | ||
| + | |width="100%"| Calculate the area of the finite region enclosed by the curves <math> y=x+2, y=1 </math> and <math> y=\frac{1}{x}</math>. | ||
| + | |- | ||
| + | |e) | ||
| + | |width="100%"| Calculate the area of the region given by the inequality, <math>x^2\le y\le x+2</math>. | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 2.1:4|Lösning a|Lösning 2.1:4a|Lösning b|Lösning 2.1:4b|Lösning c|Lösning 2.1:4c|Lösning d|Lösning 2.1:4d|Lösning e|Lösning 2.1:4e}} | ||
| + | |||
| + | ===Exercise 2.1:5=== | ||
| + | <div class="ovning"> | ||
| + | Calculate the integral | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="100%"| <math>\displaystyle \int \displaystyle\frac{dx}{\sqrt{x+9}-\sqrt{x}}\quad</math> (HINT: multiply the top and bottom by the conjugate of the denominator) | ||
| + | |- | ||
| + | |b) | ||
| + | |width="100%"| <math>\displaystyle \int \sin^2 x\ dx\quad</math> (HINT: rewrite the integrand using a trigonometric formula) | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 2.1:5|Lösning a|Lösning 2.1:5a|Lösning b|Lösning 2.1:5b}} | ||
Aktuelle Version
| Theory | Exercises |
Exercise 2.1:1
Interpret each integral as an area, and determine its value.
| a) | \displaystyle \displaystyle\int_{-1}^{2} 2\, dx | b) | \displaystyle \displaystyle\int_{0}^{1} (2x+1)\, dx |
| c) | \displaystyle \displaystyle \int_{0}^{2} (3-2x)\, dx | d) | \displaystyle \displaystyle\int_{-1}^{2}|x| \, dx |
Svar
Laden...
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 2.1:2
Calculate the integrals
| a) | \displaystyle \displaystyle\int_{0}^{2} (x^2+3x^3)\, dx | b) | \displaystyle \displaystyle\int_{-1}^{2} (x-2)(x+1)\, dx |
| c) | \displaystyle \displaystyle\int_{4}^{9} \left(\sqrt{x} - \displaystyle\frac{1}{\sqrt{x}}\right)\, dx | d) | \displaystyle \displaystyle\int_{1}^{4} \displaystyle\frac{\sqrt{x}}{x^2}\, dx |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 2.1:3
Calculate the integrals
| a) | \displaystyle \displaystyle\int \sin x\, dx | b) | \displaystyle \displaystyle\int 2\sin x \cos x\, dx |
| c) | \displaystyle \displaystyle\int e^{2x}(e^x+1)\, dx | d) | \displaystyle \displaystyle\int \displaystyle\frac{x^2+1}{x}\, dx |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 2.1:4
| a) | Calculate the area between the curve \displaystyle y=\sin x and the \displaystyle x-axis when \displaystyle 0\le x \le \frac{5\pi}{4}. |
| b) | Calculate the area under the curve \displaystyle y=-x^2+2x+2 and above the \displaystyle x-axis. |
| c) | Calculate the area of the finite region between the curves \displaystyle y=\frac{1}{4}x^2+2 and \displaystyle y=8-\frac{1}{8}x^2 (Swedish A-level 1965). |
| d) | Calculate the area of the finite region enclosed by the curves \displaystyle y=x+2, y=1 and \displaystyle y=\frac{1}{x}. |
| e) | Calculate the area of the region given by the inequality, \displaystyle x^2\le y\le x+2. |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Exercise 2.1:5
Calculate the integral
| a) | \displaystyle \displaystyle \int \displaystyle\frac{dx}{\sqrt{x+9}-\sqrt{x}}\quad (HINT: multiply the top and bottom by the conjugate of the denominator) |
| b) | \displaystyle \displaystyle \int \sin^2 x\ dx\quad (HINT: rewrite the integrand using a trigonometric formula) |
