Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ... = 1--x which lies above the axis of x. 6. A river bends around a meadow, making a curve that is approximately a parabola: y = x--4 x2, referred to a straight road that crosses the river, as axis of x; the mile is taken as the unit. How many acres of meadow are there between the road and the river? Ans. 6, nearly. 2. The Integral. In the preceding chapters we have treated the problem: Given a function; to find its derivative. The examples of the last paragraph are typical for the inverse problem: Given the derivative of a function; to find the function. Stated in equations, the problem is this. If DxU=u, or dU = udx, where u is given, to find U. The function U is called the integral of u with respect to x and is denoted as follows: Thus we have the following Definition Of An Integral. The function U is said to be the integral of u with respect to x: if DxU=u, or dU = udx. The given function u is called the integrand. For example, (1) fxdx = 1£+ G. For, if we set and differentiate this function with respect to x, we get: D„U=a. This last function is precisely the integrand, u = x6. Hence equation (1) is true hy definition. It follows from the definition that differentiation and integration are inverse processes. One undoes what the other does. Just as.y/x = x, or log e" = x, or sin (sin-1 x) = x, (2) DxJudx=u. Thus we can test conveniently any formula of integration by differentiating back. For example, is x"dx =--+C a true equation? Differentiate each side: The value of the left-hand side is, by (2), u = as". The value of the right-hand side is DZU= xn. Since these two functions are the same, i.e., since DXU= u, it follows from the definition that (3) is true. EXERCISES l. Show that 3. General Theorems. We will first show that all ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ... = 1--x which lies above the axis of x. 6. A river bends around a meadow, making a curve that is approximately a parabola: y = x--4 x2, referred to a straight road that crosses the river, as axis of x; the mile is taken as the unit. How many acres of meadow are there between the road and the river? Ans. 6, nearly. 2. The Integral. In the preceding chapters we have treated the problem: Given a function; to find its derivative. The examples of the last paragraph are typical for the inverse problem: Given the derivative of a function; to find the function. Stated in equations, the problem is this. If DxU=u, or dU = udx, where u is given, to find U. The function U is called the integral of u with respect to x and is denoted as follows: Thus we have the following Definition Of An Integral. The function U is said to be the integral of u with respect to x: if DxU=u, or dU = udx. The given function u is called the integrand. For example, (1) fxdx = 1£+ G. For, if we set and differentiate this function with respect to x, we get: D„U=a. This last function is precisely the integrand, u = x6. Hence equation (1) is true hy definition. It follows from the definition that differentiation and integration are inverse processes. One undoes what the other does. Just as.y/x = x, or log e" = x, or sin (sin-1 x) = x, (2) DxJudx=u. Thus we can test conveniently any formula of integration by differentiating back. For example, is x"dx =--+C a true equation? Differentiate each side: The value of the left-hand side is, by (2), u = as". The value of the right-hand side is DZU= xn. Since these two functions are the same, i.e., since DXU= u, it follows from the definition that (3) is true. EXERCISES l. Show that 3. General Theorems. We will first show that all ...
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