Grad Wiki - User contributions [en]

Multivariate Calculus 10B, Problem 1

2016-02-08T06:30:58Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = [\frac{1}{2}e^{2x} - \frac{1}{2 + sin(x)}e^{2x - cos(x)}]|_0^{\frac{\pi}{2}} = \frac{e^{\pi}}{6} + \frac{1}{2}(\frac{1}{e} - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T06:29:57Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|

function draw() {
var canvas = document.getElementById('canvas');
if (canvas.getContext){
var ctx = canvas.getContext('2d');

ctx.beginPath();
ctx.moveTo(75,50);
ctx.lineTo(100,75);
ctx.lineTo(100,25);

}
}
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = [\frac{1}{2}e^{2x} - \frac{1}{2 + sin(x)}e^{2x - cos(x)}]|_0^{\frac{\pi}{2}} = \frac{e^{\pi}}{6} + \frac{1}{2}(\frac{1}{e} - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T06:15:07Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-08T06:11:55Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = \frac{1}{2}e^{2x} - \frac{1}{2 + sin(x)}e^{2x - cos(x)}|_0^{\frac{\pi}{2}} = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T06:11:34Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = \frac{1}{2}e^{2x} - \frac{1}{2 + sin(x)}e^{2x - cos(x)}|_0^{\frac{\pi}{2} = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T06:02:59Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T06:02:30Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2} [e^{2x} - e^{2x - cos(x)}]~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T05:58:56Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T05:58:27Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-08T05:55:53Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution(a):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^1 \int_0^x e^{\frac{y}{x}}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

'''solution(b):'''
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!
|-
|Here we change order of integration, <math>\int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^1[xe^{\frac{y}{x}}|_{y = 0}^{y = x}]~dx = \int_0^1 x(e - 1)~dx = \frac{1}{2}x^2|_0^1(e - 1) = \frac{1}{2}(e - 1)</math>

Multivariate Calculus 10B, Problem 1

2016-02-07T09:53:08Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:52:21Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:51:36Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:50:15Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:48:42Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:48:31Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:48:20Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:48:02Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:47:32Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:46:50Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:44:54Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:44:00Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:43:12Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:42:37Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:42:10Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:37:25Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:32:31Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:30:53Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution:'''

'''a'''

:: <math>/draw [thick] (-2,2) % Draws a line
to [out=10,in=190] (2,2)
to [out=10,in=90] (6,0)
to [out=-90,in=30] (-2,-2);</math>

Multivariate Calculus 10B, Problem 1

2016-02-07T09:30:00Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution:'''

'''a'''

:: <math>/draw [thick] (-2,2) % Draws a line
to [out=10,in=190] (2,2)
to [out=10,in=90] (6,0)
to [out=-90,in=30] (-2,-2);</math>

Multivariate Calculus 10B, Problem 1

2016-02-07T09:26:51Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution:'''

'''a'''

<math>/draw [thick] (-2,2) % Draws a line
to [out=10,in=190] (2,2)
to [out=10,in=90] (6,0)
to [out=-90,in=30] (-2,-2);</math>

Multivariate Calculus 10B, Problem 1

2016-02-07T09:26:32Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution:'''

'''a'''

<math>\draw [thick] (-2,2) % Draws a line
to [out=10,in=190] (2,2)
to [out=10,in=90] (6,0)
to [out=-90,in=30] (-2,-2);</math>

Multivariate Calculus 10B, Problem 1

2016-02-07T09:25:35Z

James Ogaja 2:

Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

'''solution:'''

'''a'''

\draw [thick] (-2,2) % Draws a line
to [out=10,in=190] (2,2)
to [out=10,in=90] (6,0)
to [out=-90,in=30] (-2,-2);

Multivariate Calculus 10B, Problem 1

2016-02-07T09:22:28Z

James Ogaja 2:

Multivariate Calculus 10B, Problem 1

2016-02-07T09:20:51Z

James Ogaja 2: Created page with " Calculate the following integrals :: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math> :: b) <math>\int_0^1 \in..."

Multivariate Calculus 10B

2016-02-07T09:18:20Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]==
 Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>
:: b) <math>\int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy</math>

Multivariate Calculus 10B

2016-02-07T09:17:39Z

James Ogaja 2:

Multivariate Calculus 10B

2016-02-07T09:13:14Z

James Ogaja 2:

Multivariate Calculus 10B

2016-02-07T09:12:50Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>

Multivariate Calculus 10B

2016-02-07T09:11:40Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy</math>

Multivariate Calculus 10B

2016-02-07T09:11:29Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dx~dy</math>

Multivariate Calculus 10B

2016-02-07T09:11:04Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int _0^1 \int_y^1 e^{\frac{y}{x}}~dx</math>

Multivariate Calculus 10B

2016-02-07T09:10:33Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int _0^1 e^{\frac{y}{x}}~dx</math>

Multivariate Calculus 10B

2016-02-07T09:09:25Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int 0^1 e^{\frac{y}{x}}~dx</math>

Multivariate Calculus 10B

2016-02-07T09:09:03Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int 0^1 e^{\frac{y}{x}~dx</math>

Multivariate Calculus 10B

2016-02-07T09:08:30Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int 0^1 e^{\frac{y}{x}~dx<math>

Multivariate Calculus 10B

2016-02-07T09:08:11Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int 0^1\int y^1 e^{\frac{y}{x}~dx<math>

Multivariate Calculus 10B

2016-02-07T09:07:59Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int 0^1\int y^1 e^{\frac{y}{x}~dxdy<math>

Multivariate Calculus 10B

2016-02-07T09:06:45Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int_0^1\int_y^1 e^{\frac{y}{x}~dxdy<math>

Multivariate Calculus 10B

2016-02-07T09:06:28Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int_0^1\int_y^1 e^{\frac{y}{x}~dx~dy<math>

Multivariate Calculus 10B

2016-02-07T09:06:13Z

James Ogaja 2:

== [[Multivariate Calculus 10B, _Problem_1| Problem 1 ]]== Calculate the following integrals
:: a) <math>\int_0^1\int_y^1 e^{\frac{y}{x}~dx~dy<\math>