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2.1 Übungen

Aus Online Mathematik Brückenkurs 2

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|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).
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|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).
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|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>.
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|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>.
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Exercise 2.1:1

Interpret each integral as an area, and determine its value.

a) 2−12dx  b) 01(2x+1)dx 
c) 02(3−2x)dx  d) 2−1xdx 

Svar | Lösning a | Lösning b | Lösning c | Lösning d

Exercise 2.1:2

Calculate the integrals

a) 02(x2+3x3)dx  b) 2−1(x−2)(x+1)dx 
c) 49x−1xdx  d) 14x2xdx 

Svar | Lösning a | Lösning b | Lösning c | Lösning d

Exercise 2.1:3

Calculate the integrals

a) sinxdx  b) 2sinxcosxdx 
c) e2x(ex+1)dx  d) xx2+1dx 

Svar | Lösning a | Lösning b | Lösning c | Lösning d

Exercise 2.1:4

a) Calculate the area between the curve y=sinx and the x-axis when 0x45.
b) Calculate the area under the curve y=−x2+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)

Svar | Lösning a | Lösning b

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/2.1_%C3%9Cbungen“