Find the work along
y = x 2 2 , z = 2 3 x 3 y=\frac{x^2}{2},\space\space z=\frac{2}{3}x^3 y = 2 x 2 , z = 3 2 x 3 from x =0 to x =1:
A = ∫ L X d x + Y d y + Z d z = = ∫ 0 1 ( x y + 3 z ) d x + ( 2 y 2 − x 2 ) d y + ( z − 2 y ) d z . A=\int^LXdx+Ydy+Zdz=\\
=\int^1_0(xy+3z)dx+(2y^2 - x^2)dy+(z - 2y)dz. A = ∫ L X d x + Y d y + Z d z = = ∫ 0 1 ( x y + 3 z ) d x + ( 2 y 2 − x 2 ) d y + ( z − 2 y ) d z . From the first pair of expressions, we have
d y = x d x , d z = 2 x 2 d x . dy=xdx,\space\space dz=2x^2dx. d y = x d x , d z = 2 x 2 d x . Substitute this, as well as the first pair of equations:
A = ∫ 0 1 ( x ⋅ x 2 2 + 3 ⋅ 2 x 3 3 ) d x + + [ 2 ( x 2 2 ) 2 − x 2 ] x d x + + ( 2 x 3 3 − 2 ⋅ x 2 2 ) 2 x 2 d x = = ∫ 0 1 [ 5 x 3 2 + x 5 2 − x 3 + 4 x 5 3 − 2 x 4 ] d x = = ∫ 0 1 [ 11 x 5 6 − 2 x 4 + 3 x 3 2 ] d x = = 11 x 6 36 − 2 x 5 5 + 3 x 4 8 ∣ 0 1 = = ( 11 ⋅ 1 6 36 − 2 ⋅ 1 5 5 + 3 ⋅ 1 4 8 ) − 0 = 101 360 . A=\int^1_0(x\cdot\frac{x^2}{2}+3\cdot\frac{2x^3}{3})dx+\\ \space\\+\bigg[2\bigg(\frac{x^2}{2}\bigg)^2 - x^2\bigg]xdx+\\ \space\\
+\bigg(\frac{2x^3}{3} - 2\cdot\frac{x^2}{2}\bigg)2x^2dx=\\ \space\\
=\int^1_0\bigg[\frac{5x^3}{2}+\frac{x^5}{2}-x^3+\frac{4x^5}{3}-2x^4\bigg]dx=\\ \space\\
=\int^1_0\bigg[\frac{11x^5}{6}-2x^4+\frac{3x^3}{2}\bigg]dx=\\ \space\\
=\frac{11x^6}{36}-\frac{2x^5}{5}+\frac{3x^4}{8}\bigg|^1_0=\\
\space\\=\bigg(\frac{11\cdot1^6}{36}-\frac{2\cdot1^5}{5}+\frac{3\cdot1^4}{8}\bigg)-0=\frac{101}{360}. A = ∫ 0 1 ( x ⋅ 2 x 2 + 3 ⋅ 3 2 x 3 ) d x + + [ 2 ( 2 x 2 ) 2 − x 2 ] x d x + + ( 3 2 x 3 − 2 ⋅ 2 x 2 ) 2 x 2 d x = = ∫ 0 1 [ 2 5 x 3 + 2 x 5 − x 3 + 3 4 x 5 − 2 x 4 ] d x = = ∫ 0 1 [ 6 11 x 5 − 2 x 4 + 2 3 x 3 ] d x = = 36 11 x 6 − 5 2 x 5 + 8 3 x 4 ∣ ∣ 0 1 = = ( 36 11 ⋅ 1 6 − 5 2 ⋅ 1 5 + 8 3 ⋅ 1 4 ) − 0 = 360 101 .
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