Definite Integration
<Text-field style="Heading 1" layout="Heading 1">Problem 4-3</Text-field> JSFH Evaluate the definite integral 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 LCotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiYUdGKCEiIiomRioiIiItSSRjb3NHRiVGKUYtRi0tRiQ2I0kiYkdGKEYtKiZGMkYtLUYvRjFGLUYr (a) directly in Maple. (b) by applying the definition of the definite integral.
JSFH
<Text-field style="Heading 1" layout="Heading 1">Evaluation of the Integral</Text-field> JSFH Control-drag the integral. Context Menu: Evaluate and Display Inline JSFH JSFHJSFH Two ways to construct the definite integral Expression palette: Definite integral template JSFHJSFH Enter integrand. Context Menu: Constructions\342\211\273Definite Integral Context Menu: Evaluate (from inert) JSFH JSFH JSFH
JSFH JSFH
<Text-field style="Heading 1" layout="Heading 1">Definite Integral by Definition</Text-field> JSFH Define the integrand as a function: write 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 Context Menu: Assign Function JSFH JSFH Tools\342\211\273Tasks\342\211\273Browse: Calculus - Integral\342\211\273Integration\342\211\273Riemann Sums\342\211\273Left Type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and press the Enter key. Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ as the interval Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as the symbolic number of partitions
The Left Riemann Sum Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: 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 LCYqJClJJl9feF9fRzYiIiIjIiIiRihGKEYo Enter the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+: 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 NyQiIiEiIiI= Enter the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiQtRiM2JS1JI21uR0YkNidRIjRGJy8lK2ZvcmVncm91bmRHUStbOTUsMCwyMTddRicvJSxwbGFjZWhvbGRlckdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdROE1hcGxlfklucHV0flBsYWNlaG9sZGVyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGQS8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGREZB IiIl The left Riemann sum: 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 LCQqJiMiIiIiIiVGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJiNGJSIjO0YlKUkiaUdGLCIiI0YlRiVGJUYlL0YzOyIiISIiJEYlRiU= Value of the Riemann sum: 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 IyIjUiIjSw==
JSFH Expression palette: Limit operator (Reference sum by equation label.) Press the Enter key. JSFH JSFHJSFH JSFH
<Text-field style="Heading 1" layout="Heading 1">Solution from First Principles</Text-field> JSFH Define the integrand as a function: write 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 Context Menu: Assign Function JSFH Enter the stepsize 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 Context Menu: Assign to a Name\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= JSFH Enter the left Riemann sum 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 Expression palette: Limit operator Expression palette: Summation symbol (Sigma) Press the Enter key. JSFH
JSFH
JSFH JSFH JSFH ; JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH