Question

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Answer 1

Question :

If $\sf \bar{F}=2z \hat{i}-x\hat{j}+y \hat{k}$ , evaluate $\displaystyle \sf \int \int \int_{V} \bar{F}\ dv$ , where v is the region bounded by the surfaces x = 0 , y = 0 , x = 2 , y = 4 , z = x² , z = 2 .

Solution :

$\displaystyle \sf \int \int \int_{V} \bar{F}\ dv$

$\longrightarrow \displaystyle \sf \int_{0}^{2}dx \int_{0}^{4}dy \int_{x^2}^{2}(2z \hat{i}- x \hat{j}+y \hat{k})\ dz$

$\longrightarrow \displaystyle \sf \int_{0}^{2}dx \int_{0}^{4}dy \left[ (z^2 \hat{i}- xz \hat{j}+yz \hat{k}) \right]_{x^2}^{2}$

$\\ \longrightarrow \displaystyle \sf \int_{0}^{2}dx \int_{0}^{4}dy \left[ (4 \hat{i}- 2x \hat{j}+2y \hat{k} - x^4 \hat{i}+ x^3 \hat{j} - x^2 y \hat{k}) \right]$

$\\ \longrightarrow \displaystyle \sf \int_{0}^{2}dx \left[ (4y \hat{i}- 2xy \hat{j}+y^2 \hat{k} - x^4y \hat{i}+ x^3y \hat{j} - \frac{x^2 y^2}{2}\ \hat{k}) \right]_{0}^{4}$

$\\ \longrightarrow \displaystyle \sf \int_{0}^{2}dx \left[ (16 \hat{i}- 8x \hat{j}+16 \hat{k} - 4x^4 \hat{i}+ 4x^3 \hat{j} - \frac{16x^2 }{2}\ \hat{k}) \right]$

$\\ \longrightarrow \displaystyle \sf \left[ \left( 16x \hat{i}- 4x^2 \hat{j}+16x \hat{k} - \frac{4x^5}{5} \hat{i}+ x^4 \hat{j} - \frac{8x^3 }{3}\ \hat{k} \right) \right]_{0}^{2}$

$\\ \longrightarrow \displaystyle \sf \left(32 \hat{i}- 16 \hat{j}+32 \hat{k} - \frac{128}{5} \hat{i}+ 16 \hat{j} - \frac{64 }{3}\ \hat{k} \right)$

$\longrightarrow \displaystyle \sf \left( \dfrac{32}{5} \hat{i}+ \frac{32}{3}\ \hat{k} \right)$

$\longrightarrow \displaystyle \sf \pink{\dfrac{32}{15} \left( 3 \hat{i}+5 \hat{k} \right)}$