Math  /  Calculus

Question16. Express Exz3dV\iiint_{E} x z^{3} d V, where E={(x,y,z):0x1,0yx,0zx2y2}E=\left\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq x, 0 \leq z \leq x^{2} y^{2}\right\} in a form that can be evaluated. A. 0x2y20x01xz3dxdydz\int_{0}^{x^{2} y^{2}} \int_{0}^{x} \int_{0}^{1} x z^{3} d x d y d z B. 0x010x2y2xz2dzdxdy\int_{0}^{x} \int_{0}^{1} \int_{0}^{x^{2} y^{2}} x z^{2} d z d x d y C. 010x2y20xxz2dydzdx\int_{0}^{1} \int_{0}^{x^{2} y^{2}} \int_{0}^{x} x z^{2} d y d z d x D. 010x0x2y2xz2dzdydx\int_{0}^{1} \int_{0}^{x} \int_{0}^{x^{2} y^{2}} x z^{2} d z d y d x

Studdy Solution
Express the triple integral in the form:
010x0x2y2xz3dzdydx \int_{0}^{1} \int_{0}^{x} \int_{0}^{x^2 y^2} x z^3 \, dz \, dy \, dx
This matches option D, where the integration order is dzdydx dz \, dy \, dx and the limits correspond to the region E E .
The correct expression is:
010x0x2y2xz3dzdydx \boxed{\int_{0}^{1} \int_{0}^{x} \int_{0}^{x^2 y^2} x z^3 \, dz \, dy \, dx}

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