Antidifferentiation as Area under a CurveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 4-2</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Use a Riemann sum to find the area bounded by the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, the lines 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 and the graph of the curve corresponding to 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.
Enter (or Control-drag) the equation 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Context Menu: Assign FunctionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Tools\342\211\273Tasks\342\211\273Browse: Calculus - Integral\342\211\273Integration\342\211\273Riemann Sums\342\211\273LeftType LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and press the Enter keyEnter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ as the intervalEnter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as the symbolic number of partitions
The Left Riemann SumEnter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: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Enter the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+: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Enter the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiQtRiM2JS1JI21uR0YkNidRIjRGJy8lK2ZvcmVncm91bmRHUStbOTUsMCwyMTddRicvJSxwbGFjZWhvbGRlckdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdROE1hcGxlfklucHV0flBsYWNlaG9sZGVyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGQS8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGREZBThe left Riemann sum: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Value of the Riemann sum: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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Expression palette: Limit operator
(Reference sum by equation label.)
Press the Enter key.
Context Menu: Expand\342\211\273ExpandLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Note that if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is an antiderivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, then the area computed is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImJGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSgmbWludXM7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZURistRjY2JC1GIzYkLUYsNiVRImFGJ0YvRjJGPUY9Rj0=, and this is the content of the Fundamental Theorem of Calculus.