標題:
integration proof
以下系個問題: http://i407.photobucket.com/albums/pp158/ah-tee/int-proof.jpg
最佳解答:
Question: Suppose f is increasing and continuous on [a,b]. Let Pn denote the partition on [a,b] into n equal subintervals. Show that Lf(Pn) and Uf(Pn) both converge to 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx%20%5Ctext%7B%20as%20%7D%20n%5Crightarrow%20%5Cinfty , that is, 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%3D%20%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx%20%3D%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7DU_f(P_n) . Solution: Since f is continuous on [a,b], the integral 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx exists. Observe that 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=L_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7Df(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a))%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cint_%7Ba%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%7D%5E%7Ba%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a)%7Df(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a))dx%20%5Cle%5Cint_a%5Ebf(x)dx , 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=U_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7Df(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cint_%7Ba%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%7D%5E%7Ba%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a)%7Df(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))dx%20%5Cge%5Cint_a%5Ebf(x)dx , and 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=U_f(P_n)%20-%20L_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7D%5Bf(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))%20-%20f(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%5D%20%3D%20%5Cfrac%7Bf(b)-f(a)%7D%7Bn%7D . So we have (1) 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=L_f(P_n)%20%5Cle%20%5Cint_a%5Ebf(x)dx%20%5Cle%20U_f(P_n) , and (2) 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5BU_f(P_n)-L_f(P_n)%5D%20%3D%200 . Hence, we have 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_a%5Ebf(x)dx%20-%20%5BU_f(P_n)-L_f(P_n)%5D%20%3D%20L_f(P_n)-%5BU_f(P_n)-%5Cint_a%5Ebf(x)dx%5D%20%5Cle%20L_f(P_n)%20%5Cle%20%5Cint_a%5Ebf(x)dx . By sandwich theorem, we get 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%3D%20%5Cint_a%5Ebf(x)dx . So 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DU_f(P_n)%20%3D%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%2B%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5BU_f(P_n)-L_f(P_n)%5D%20%3D%20%5Cint_a%5Ebf(x)dx . This completes the proof.
其他解答:
integration proof
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發問:以下系個問題: http://i407.photobucket.com/albums/pp158/ah-tee/int-proof.jpg
最佳解答:
Question: Suppose f is increasing and continuous on [a,b]. Let Pn denote the partition on [a,b] into n equal subintervals. Show that Lf(Pn) and Uf(Pn) both converge to 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx%20%5Ctext%7B%20as%20%7D%20n%5Crightarrow%20%5Cinfty , that is, 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%3D%20%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx%20%3D%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7DU_f(P_n) . Solution: Since f is continuous on [a,b], the integral 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_%7Ba%7D%5E%7Bb%7Df(x)dx exists. Observe that 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=L_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7Df(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a))%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cint_%7Ba%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%7D%5E%7Ba%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a)%7Df(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a))dx%20%5Cle%5Cint_a%5Ebf(x)dx , 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=U_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7Df(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cint_%7Ba%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%7D%5E%7Ba%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a)%7Df(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))dx%20%5Cge%5Cint_a%5Ebf(x)dx , and 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=U_f(P_n)%20-%20L_f(P_n)%20%3D%20%5Csum_%7Bi%3D1%7D%5En%5Cfrac%7B1%7D%7Bn%7D%5Bf(a%2B%5Cfrac%7Bi%7D%7Bn%7D(b-a))%20-%20f(a%2B%5Cfrac%7Bi-1%7D%7Bn%7D(b-a)%5D%20%3D%20%5Cfrac%7Bf(b)-f(a)%7D%7Bn%7D . So we have (1) 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=L_f(P_n)%20%5Cle%20%5Cint_a%5Ebf(x)dx%20%5Cle%20U_f(P_n) , and (2) 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5BU_f(P_n)-L_f(P_n)%5D%20%3D%200 . Hence, we have 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cint_a%5Ebf(x)dx%20-%20%5BU_f(P_n)-L_f(P_n)%5D%20%3D%20L_f(P_n)-%5BU_f(P_n)-%5Cint_a%5Ebf(x)dx%5D%20%5Cle%20L_f(P_n)%20%5Cle%20%5Cint_a%5Ebf(x)dx . By sandwich theorem, we get 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%3D%20%5Cint_a%5Ebf(x)dx . So 圖片參考：http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Clim_%7Bn%5Crightarrow%5Cinfty%7DU_f(P_n)%20%3D%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7DL_f(P_n)%20%2B%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5BU_f(P_n)-L_f(P_n)%5D%20%3D%20%5Cint_a%5Ebf(x)dx . This completes the proof.
其他解答:
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