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Average Value Theorem
| Definition. Let f be a function which is continuous on the closed interval [a, b]. The average value of f from x = a to x = b is the integral
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1-Dimensional Motion Problems:
Integrals can be used to help solve these one dimensional motion problems. As discussed before, there are three variants to the 1-Dimensional Motion Problems, which are Position, Velocity, Accelaration. The derivative of the Position function will give you the Velocity function, and the derivative of the Velocity function will give you the Accelaration function. Integration comes into play when you are given the Accelaration and are asked to evaluate the problem with the Velocity or even Position function.
Differential Equations
Introduction. A useful tool in the study of indefinite integrals (and differential equations) is a slope field (or direction field). Using slope fields, one can find an antiderivative both graphically and numerically.
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Definition. Given a function y = f(x), a slope field is drawn for f by choosing a collection of evenly-spaced points x1, x2, ..., xm along the x-axis and a collection of evenly-spaced points y1, y2, ..., yn along the y-axis. At each point (xi, yj), a small line with slope f(xi) is drawn. | |
By specifying one point (a, b), we can find an antiderivatve F of f such that b = F(a). If f is continuous then there is only one such antiderivative. By using a slope field, you can get an idea of what the graph of an antiderivative of f looks like.
This is the slope field for f(x) = sin(x)
Volume of Solids in Revolution:
This following animation shows the effects of being rotated around the X-Axis:
| Theorem. Let f be a function which is continuous on the closed interval [a, b]. The volume of the solid obtained by rotating the graph of f from x = a to x = b about the y-axis is the integral
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