Calculus Archive: Questions from November 24, 2022
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2 answers
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Transform the equation sin x sin y dx + cos x cos y dy = 0 to a variable separable form f(x) dx + g (y) dy = 0
Transform the equation \[ \sin x \sin y d x+\cos x \cos y d y=0 \] to a variable separable form \[ f(x) d x+g(y) d y=0 \] \[ \frac{\sin x d x}{\cos x}+\frac{\cos y d y}{\sin y}=0 \] \[ \sin x \cos x d2 answers -
Evaluar las siguientes integrales: 1. \( \int \ln x^{3} d x \) 2. \( \int\left(t^{3}-2 t^{2}+4 t-3\right) e^{2 t} d t \) 3. \( \int \sin ^{3} x \cos ^{2} x d x \) 4. \( \int \sec ^{3} \frac{x}{2} \tan1 answer -
Consider the following. \[ f(x, y, z)=y^{2} e^{x y z}, \quad P(0,2,-2), \quad \mathbf{u}=\left\langle\frac{3}{13}, \frac{4}{13}, \frac{12}{13}\right\rangle \] (a) Find the gradient of \( f \). \[ \nab2 answers -
Evaluate the triple integral \( \iiint_{Q} f(x, y, z) d V \). \[ \begin{array}{l} f(x, y, z)=7 x+9 y-7 z, Q=\{(x, y, z) \mid 0 \leq x \leq 8,-5 \leq y \leq 5,0 \leq z \leq 8\} \\ \iiint_{Q} f(x, y, z)2 answers -
Evaluate the triple integral \( \iiint_{Q} f(x, y, z) d V \). \[ \begin{array}{l} f(x, y, z)=7 x+9 y-7 z, Q=\{(x, y, z) \mid 0 \leq x \leq 8,-5 \leq y \leq 5,0 \leq z \leq 8\} \\ \iiint_{Q} f(x, y, z)2 answers -
tion \( \left(D^{2}-3 D+2\right) y=e^{x} \) \( y=c_{1} e^{2 x}+c_{2} e^{-3 x}-x e^{x} \) \( y=c_{1} e^{-x}+c_{2} e^{-2 x}-x e^{x} \) \( y=c_{1} e^{x}+c_{2} e^{2 x}-x e^{x} \) \( y=c_{1} e^{x}+c_{2} e^2 answers -
Evaluate the triple integral \( \iiint_{Q} f(x, y, z) d V \). \[ \begin{array}{l} f(x, y, z)=7 x+3 y-5 z, Q=\{(x, y, z) \mid 0 \leq x \leq 5,-4 \leq y \leq 4,1 \leq z \leq 4\} \\ \iiint_{Q} f(x, y, z)2 answers -
Evaluate \( \iiint_{E} f(x, y, z) d V \) for the specified function \( f \) and \( \mathcal{B} \) : \[ f(x, y, z)=\frac{z}{x} \quad 3 \leq x \leq 9,0 \leq y \leq 9,0 \leq z \leq 6 \] \[ \iiint_{5} f(x2 answers -
1. The flow curves of F4(x,y)=xi+yj+zk are rays emanating from the origin. 2. 2. The domain of F3(x,y) is in all the plane. 3. The flow curves of F1 are rays emanating from the origin 4. The flo
\[ \begin{array}{l} F_{1}(x, y)=x \mathbf{i}+\mathbf{y} \mathbf{j} ; \quad \mathbf{F}_{2}(\mathbf{x}, \mathbf{y})=-\mathbf{y} \mathbf{i}+\mathbf{x} \mathbf{j} \\ F_{3}(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}2 answers -
\( \int_{-1}^{0} \frac{8 x}{\left(4 x^{2}+1\right)^{2}} d x ; u=4 x^{2}+1 \) 10) \( \int_{0}^{1}-12 x^{2}\left(4 x^{3}-1\right)^{3} d x ; u=4 x^{3}-1 \) \( \int_{-1}^{2} 6 x\left(x^{2}-1\right)^{2} d2 answers -
Evaluate \( \iiint_{B} f(x, y, z) d V \) for the specified function \( f \) and \( B \) : \[ f(x, y, z)=\frac{z}{x} \quad 3 \leq x \leq 6,0 \leq y \leq 9,0 \leq z \leq 4 \]2 answers -
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9-18 Evaluate the triple integral. 9. \( \iiint_{E} y d V \), where \[ E=\{(x, y, z) \mid 0 \leqslant x \leqslant 3,0 \leqslant y \leqslant x, x-y \leqslant z \leqslant x+y\} \] 10. \( \iiint_{E} e^{z2 answers -
Rewrite the triple integral \( \int_{0}^{1} \int_{0}^{x} \int_{0}^{y} f(x, y, z) d z d y d x \) as \( \int_{a}^{b} \int_{a_{1}(z)}^{g_{2}(z)} \int_{h_{1}(u, x)}^{h_{2}(y, z)} f(x, y, z) d x d y d z \)2 answers -
Express the integral \( \iiint_{E} f(x, y, z) d V \) as an iterated integral in six different ways, where \( \mathrm{E} \) is the solid bounded by \( z=0, x=0, z=y-5 x \) and \( y=10 \). \[ \begin{arr1 answer -
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Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For \[ y=3-z^{2}, \quad 0 \leq x, z \leq 9 ; \quad f(x, y, z)=z \] \[ \iint_{\mathcal{S}} f(x, y, z) d S= \]2 answers